A377964
Expansion of e.g.f. (1+x) * exp(x*(1+x)^3).
Original entry on oeis.org
1, 2, 9, 58, 389, 3186, 29437, 294554, 3233673, 38350594, 484794641, 6522118362, 92857444429, 1390937221298, 21858658599429, 359271578140666, 6156249977141777, 109722278546645634, 2029772196329985433, 38893956306343711994, 770622936760496106261, 15763542538016019828082
Offset: 0
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a(n, s=1, t=3) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);
A377965
Expansion of e.g.f. (1+x)^2 * exp(x*(1+x)^2).
Original entry on oeis.org
1, 3, 11, 55, 309, 1931, 13543, 101991, 828425, 7192819, 66002691, 639830423, 6510397501, 69266297595, 768989536799, 8876171274631, 106301772962193, 1318277355041891, 16892429768517115, 223330116792810999, 3041570471301007301, 42611228176879105003
Offset: 0
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a(n, s=2, t=2) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);
A377967
Expansion of e.g.f. (1+x)^3 * exp(x*(1+x)^3).
Original entry on oeis.org
1, 4, 19, 124, 961, 8236, 79339, 840484, 9595009, 117764596, 1542837091, 21406165804, 313381177729, 4822681240924, 77704955681851, 1307128152596116, 22899018541506049, 416756647023727204, 7863586717014612019, 153550319029835965276, 3097694623619639050561
Offset: 0
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With[{nn=20},CoefficientList[Series[(1+x)^3 Exp[x*(1+x)^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2025 *)
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a(n, s=3, t=3) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);
A378017
E.g.f. satisfies A(x) = (1+x)^2 * exp( x * (1+x) * A(x) ).
Original entry on oeis.org
1, 3, 15, 148, 2077, 38326, 883369, 24431436, 789531705, 29213730010, 1218401262541, 56566561281616, 2893886178904549, 161768999193007974, 9811122121462081281, 641683497800057913556, 45022670799746182036465, 3373421864644139722767538, 268843153008388446079159573
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)^2*exp(-lambertw(-x*(1+x)^3))))
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a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(3*k+2, n-k)/k!);
Showing 1-4 of 4 results.