A356812
Expansion of e.g.f. exp(x * (1 - exp(2*x))).
Original entry on oeis.org
1, 0, -4, -12, 16, 400, 2208, -448, -131840, -1357056, -4820480, 71120896, 1537308672, 14006460416, 3075702784, -2224350781440, -41354996154368, -359660395495424, 1675436608585728, 121894823709900800, 2317859245604208640, 20543311167964053504
Offset: 0
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With[{nn=30},CoefficientList[Series[Exp[x(1-Exp[2x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 04 2023 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-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+1)*x)^(k+1)))
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a(n) = sum(k=0, n, (-1)^k*(2*k+1)^(n-k)*binomial(n, k));
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a(n) = n!*sum(k=0, n\2, (-1)^k*2^(n-k)*stirling(n-k, k, 2)/(n-k)!);
A356816
Expansion of e.g.f. exp(-x * (exp(3*x) + 1)).
Original entry on oeis.org
1, -2, -2, 1, 88, 583, 676, -35597, -519392, -3359393, 19013884, 896435395, 13640180896, 85591357135, -1527872118356, -61100053650053, -1076294742932288, -7610985095240513, 200631806070276988, 9284475508083767059, 200226297062313730816, 1940767272243466116463
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(exp(3*x)+1))))
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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));
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)!);
A356820
Expansion of e.g.f. exp(-x * exp(3*x)).
Original entry on oeis.org
1, -1, -5, -10, 73, 1004, 5473, -15562, -746447, -9174088, -41916959, 823985546, 24629093641, 335144105828, 1248594602305, -67564407472426, -2160461588461343, -34957074099518608, -154556217713939903, 10500560586914149250, 409146670525578079801
Offset: 0
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With[{nn=20},CoefficientList[Series[Exp[-x Exp[3x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 13 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*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*x)^(k+1)))
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a(n) = sum(k=0, n, (-1)^k*(3*k)^(n-k)*binomial(n, k));
Showing 1-4 of 4 results.