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));
A356818
Expansion of e.g.f. exp(-x * (exp(x) + 1)).
Original entry on oeis.org
1, -2, 2, 1, 0, -17, -32, 103, 976, 2287, -12816, -143585, -481016, 2339335, 39769720, 209863327, -397553376, -16949434913, -142681662368, -233212601153, 9138353475736, 128343346833463, 702261255539496, -4251314594919617, -135331386127555856
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(exp(x)+1))))
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(k-1)*x)^(k+1)))
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a(n) = sum(k=0, n, (-1)^k*(k-1)^(n-k)*binomial(n, k));
A356817
a(n) = Sum_{k=0..n} (-1)^k * (k*n-1)^(n-k) * binomial(n,k).
Original entry on oeis.org
1, -2, 0, 1, 144, 4143, 110368, 2535475, 13299968, -5169863825, -639341093376, -59073970497885, -4677854594527232, -276406098219258425, 2399871442122924032, 5163244810691492730907, 1331213942683118587674624, 262517264591996332314037215
Offset: 0
Showing 1-4 of 4 results.