A356813
Expansion of e.g.f. exp(x * (1 - exp(3*x))).
Original entry on oeis.org
1, 0, -6, -27, 0, 1215, 12312, 45927, -657072, -15857937, -167699160, -266960529, 29356170984, 700068823623, 8419188469104, -1491045413265, -2856006296224992, -79065447339366945, -1162293393139510824, -744123842820101745, 538503788896323210360
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-exp(3*x)))))
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(3*k+1)*x)^(k+1)))
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a(n) = sum(k=0, n, (-1)^k*(3*k+1)^(n-k)*binomial(n, k));
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a(n) = n!*sum(k=0, n\2, (-1)^k*3^(n-k)*stirling(n-k, k, 2)/(n-k)!);
A356815
Expansion of e.g.f. exp(-x * (exp(2*x) + 1)).
Original entry on oeis.org
1, -2, 0, 4, 32, 48, -608, -6400, -24064, 163072, 3567104, 28394496, 6535168, -3250745344, -50725740544, -344530853888, 2476610551808, 110057610608640, 1655672654135296, 9616664975114240, -195178079811272704, -6998474114188967936, -110894925369151848448
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(exp(2*x)+1))))
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(2*k-1)*x)^(k+1)))
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a(n) = sum(k=0, n, (-1)^k*(2*k-1)^(n-k)*binomial(n, k));
A356814
a(n) = Sum_{k=0..n} (-1)^k * (k*n+1)^(n-k) * binomial(n,k).
Original entry on oeis.org
1, 0, -4, -27, -64, 4375, 199584, 6739607, 169934848, -1012395105, -709624000000, -86599643309201, -8221227668471808, -638169258399740977, -27617164284655812608, 3853095093357099609375, 1568756883209662050074624, 360407172063462944082773311
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*(k*n+1)^(n-k)*binomial(n, k));
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a(n) = n!*sum(k=0, n\2, (-1)^k*n^(n-k)*stirling(n-k, k, 2)/(n-k)!);
A356819
Expansion of e.g.f. exp(-x * exp(2*x)).
Original entry on oeis.org
1, -1, -3, -1, 41, 239, 229, -8401, -87151, -324577, 3238541, 70271519, 601086265, 142860431, -81504662539, -1393683935281, -10777424809951, 63537986981183, 3552608426329117, 60283510555017023, 441644419610814281, -6191820436867600081
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*exp(2*x))))
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-2*k*x)^(k+1)))
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a(n) = sum(k=0, n, (-1)^k*(2*k)^(n-k)*binomial(n, k));
Showing 1-4 of 4 results.