When Ted Page became a grandfather in 2014, he wanted to hear other men’s stories, “real experiences that hopefully would guide newcomers,” he recalled.

“We’re shifting into a different part of our lives,” said Mr. Page, a 63-year-old from Lexington, Mass. “I was trying to picture what it would be like.”

Mr. Page, a co-founder of a marketing consulting firm, wondered whether it was wise to pass around photos of his family, including his grandchildren, at meals with clients. “There’s a bit of a stigma,” he said. “It could be seen as an indication that one is ready for the pasture.”

As he searched the web for information about this new role, he found a dozen blogs where grandmothers were chatting, seeking advice and exchanging ideas. But blogs for grandfathers? None. So he started his own site, Good Grandpa, where he ruminates about things like his own grandfather’s wisdom and the height markings on a kitchen wall that’s about to undergo renovation.

Greg Payne, 53, an I.T. project manager in suburban Atlanta, encountered the same void when his first grandchild was born four years ago. After struggling to find books about grandfathering, he turned to movies and television but came up empty-handed, apart from the dispiriting 2013 movie “Bad Grandpa,” featuring a lecherous alcoholic who neglects an 8-year-old.

In popular culture, “grandfathers typically are checked out,” Mr. Payne said. He started the “Cool Grandpa” podcast in 2020, because “there is so much more to say about the relationship.”

Grandfathers are often excluded from societal narratives about relationships between generations, somehow secondary to grandmothers. That’s hard to quantify, because there’s limited recent research about noncustodial grandparents — though AARP conducts regular surveys — and grandfathering seems particularly unexamined. Factors like retirement status, health and geographic distance help determine men’s involvement in their grandchildren’s lives, but the cultural expectations for how grandfathers should behave are hazy.

“You get these gendered stereotypes,” said Robin Mann, a sociologist at Bangor University in Wales who studies the way men in England approach grandparenting and how it relates to masculinity later in life. “Grandfathers themselves often see the role as feminized, a woman’s role.”

In suburban Philadelphia, for instance, Alice Linder, 78, described her husband’s unwillingness to sit on the floor and play with their 5-year-old grandson, who lives nearby.

“He can’t stand the noise,” said Ms. Linder, who spends several afternoons a week caring for her grandson. Her husband was not a very hands-on father himself, she said. “His method of being a grandparent is to stay away,” she added.

There are, of course, many engaged and enthusiastic grandfathers. Some men, like George Schweitzer, a retired media executive in New York, view the role as a second chance: When his three daughters were young, Mr. Schweitzer, now 71, was ascending the corporate ladder. He made it a point to come home in time for their evening routines, but “I’d lie on their beds to read to them and fall asleep before they did,” he said. Now, Mr. Schweitzer and his wife, who have five grandchildren, are “as involved as we can be without being obnoxious.”

Other men provide regular care for their grandchildren — what sociologists call “intensive grandparenting” — to support their adult children’s economic or career needs. And some grandfathers just want to participate meaningfully in their expanding families.

Barry Sage-El, 69, a retired software designer in Montclair, N.J., describes himself as “the master of the sleepover — and the sleepover breakfast.” Once or twice a month, his three grandchildren, ages 3 to 5, take over the sofa bed in his second bedroom — a fun event, but also a break for their parents.

Because his wife still works, Mr. Sage-El is the one who usually oversees their walks to the park and the gelato store, the children’s art projects, and the pancakes they make together the following morning. “We just like to see them grow,” he said. “I’m not only their friend, but an influencer who can help shape them with things they’ll remember.”

For Jonathan Wolf, 64, the impetus for his involvement was simple: His grandson, Nathan, was born just over a month before the coronavirus pandemic started. Nathan’s working parents were uneasy, after their parental leaves ended, about sending him to day care. Mr. Wolf, a retired high school physics teacher in Wanaque, N.J., called his response “instinctual and automatic.”

“I’m not working,” he said. “If they need me to help, I’ll help.”

He watches Nathan five days a week, often for eight to 10 hours each day. Now that Nathan is 3 and attends preschool in the afternoons, Mr. Wolf can fit in a nap — for himself. “Running after a 2- or 3-year-old is exhausting,” he said. But he plans to continue, even when Nathan goes to school full time.

Overall, grandmothers still take the lead in spending time with grandchildren, often rearranging their schedules to do so, said Madonna Harrington Meyer, a sociologist at Syracuse University and the author of “Grandmothers at Work: Juggling Families and Jobs.”

And Jennifer Utrata, a sociologist at the University of Puget Sound who has interviewed dozens of parents and grandparents, found that even when grandfathers are involved, “the care is often arranged, monitored and checked on by the grandmothers.” The grandfathers see their role as supplementary, helping their spouses out. “Most intensive grandparenting is still intensive grandmothering,” Dr. Utrata said.

Researchers, however, believe change is on the horizon. Cultural and demographic trends, including better health and longer lives, mean that grandfathers can take more active roles. And there’s some evidence that American fathers spend considerably more time caring for children than their predecessors did: an average of eight hours a week in 2016, compared with just 2.5 hours in 1965, according to the Pew Research Center. As contemporary dads become grandpas, caring for kids may feel satisfying and familiar.

“I’m seeing that both grandmothers and grandfathers want to be involved, though I don’t think grandfathers are changing diapers as much,” said Kathy Hirsh-Pasek, a developmental psychologist at Temple University and a senior fellow at the Brookings Institution.

Still, there are many ways men can continue to develop closer bonds with their grandchildren, she noted. A few ways to get started:

Ask your adult children for help.

Let them know these connections are important to you. Parents can facilitate them by arranging FaceTime and Zoom sessions between grandfathers, especially those far away, and the kids.

But strained relationships might require a little negotiation. “Not-so-great fathers sometimes make very good grandfathers,” said Karl Pillemer, a psychologist at Cornell University and the author of “Fault Lines: Fractured Families and How to Mend Them.”

Grandfathers may need to acknowledge their own parental shortcomings, making clear what will be different this time; they should also discuss and accept the terms and boundaries their adult children set. Dr. Pillemer suggested asking, “How can I be helpful to you?”

Share interests with your grandchildren.

Dr. Hirsh-Pasek urged parents to share information about their children’s current obsessions, so grandfathers can bone up on dinosaur facts or watch a few episodes of “Bluey” and talk with their grandkids about them. For online visits, “find books they can read together and things they can do together,” she said.

She also suggested that grandfathers share their own interests. Usually, “if you like woodworking or boating, kids will enjoy learning about it,” she said. Sports are another area where many grandfathers seem happy to get involved, Dr. Mann said.

Schedule one-on-one time.

Dr. Hirsh-Pasek advised grandfathers in heterosexual relationships to spend some time with their grandchildren on their own, apart from their spouses. Otherwise, they might defer to their wives, especially if “she is the child-raiser, the more experienced one,” she said.

Men who embrace grandfathering say the role comes with immense rewards. Mr. Schweitzer and his wife see three of their grandkids who live nearby in Manhattan several times a week; they travel as often as possible to Los Angeles to visit two other grandchildren, relying on FaceTime in between. In the summer, they host the whole family in a rented house for a week, complete with Cousin Camp T-shirts.

The result? “Sheer joy, pride, fulfillment, purpose,” Mr. Schweitzer said. “It’s great to see kids grow up, and to be a part of it.”