A354242
Expansion of e.g.f. 1/sqrt(5 - 4 * exp(x)).
Original entry on oeis.org
1, 2, 14, 158, 2486, 50222, 1239254, 36126638, 1214933846, 46299580142, 1971815255894, 92809525295918, 4784166929982806, 268050260650705262, 16219498558371118934, 1054102762745609325998, 73229184033780135425366, 5415407651703010175897582
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(5-4*exp(x))))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(exp(x)-1)^k)))
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a(n) = sum(k=0, n, (2*k)!*stirling(n, k, 2)/k!);
A308490
a(0) = 1, a(n) = Sum_{k=1..n} stirling2(n,k) * k^(2*k).
Original entry on oeis.org
1, 1, 17, 778, 70023, 10439451, 2327592658, 725325847443, 301054612941037, 160546901676583432, 106969402879501806589, 87079496403914056543799, 85043317211453886535179728, 98135961356804028347727824541, 132097548629285541942722646521053
Offset: 0
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Join[{1}, Table[Sum[k^(2*k)*StirlingS2[n, k], {k, 1, n}], {n, 1, 20}]]
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*(exp(x)-1))^k/k!))) \\ Seiichi Manyama, Feb 04 2022
A323280
a(n) = Sum_{k=0..n} binomial(n,k) * k^(2*k).
Original entry on oeis.org
1, 2, 19, 781, 68553, 10100761, 2236373953, 693667946945, 286962262702657, 152652510206521921, 101513694573289791441, 82511051259976074269425, 80480313356721971865934369, 92773167329045961244649105633, 124768226258051318899374299271601
Offset: 0
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Table[1 + Sum[Binomial[n, k]*k^(2*k), {k, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 31 2019 *)
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a(n) = sum(k=0, n, binomial(n, k)*k^(2*k));
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k^2*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022
A316748
Stirling transform of (3*n)!.
Original entry on oeis.org
1, 6, 726, 365046, 481183926, 1312473466806, 6422019989033526, 51225575261701080246, 621880652519326246083126, 10911229213845806303174823606, 265743324574322126992546955062326, 8697919110119969555113124407898635446, 372566878251517048881238923757823056246326
Offset: 0
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Table[Sum[StirlingS2[n, k]*(3*k)!, {k, 0, n}], {n, 0, 15}]
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k)!*(exp(x)-1)^k/k!))) \\ Seiichi Manyama, May 21 2022
A354244
Expansion of e.g.f. Sum_{k>=0} (2*k)! * (-log(1-x))^k / k!.
Original entry on oeis.org
1, 2, 26, 796, 44916, 4058448, 537029616, 97903213056, 23525415709632, 7205450503530816, 2740066802232081984, 1266655419369548369280, 699532666466320784246400, 454880976674201215672273920, 344008843780994236543882521600
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k)!*(-log(1-x))^k/k!)))
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a(n) = sum(k=0, n, (2*k)!*abs(stirling(n, k, 1)));
A354243
Expansion of e.g.f. Sum_{k>=0} (2*k)! * log(1+x)^k / k!.
Original entry on oeis.org
1, 2, 22, 652, 36252, 3249648, 427841136, 77725790784, 18629187576192, 5694658698037824, 2162203542669622464, 998275836346954738560, 550745779092109449586560, 357819370067278253918223360, 270404811566689476740771496960
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k)!*log(1+x)^k/k!)))
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a(n) = sum(k=0, n, (2*k)!*stirling(n, k, 1));
Showing 1-6 of 6 results.
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