A354311
Expansion of e.g.f. exp( x/2 * (exp(2 * x) - 1) ).
Original entry on oeis.org
1, 0, 2, 6, 28, 160, 1056, 7784, 63568, 569664, 5542240, 58038112, 650045760, 7746901760, 97790608384, 1302349549440, 18235836899584, 267663541270528, 4107395264113152, 65739857693144576, 1095095457262013440, 18949711553467957248, 340036076121127395328
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/2*(exp(2*x)-1))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*2^(j-2)*binomial(i-1, j-1)*v[i-j+1])); v;
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a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);
A354325
Expansion of e.g.f. 1/(1 - x/4 * (exp(2 * x) - 1)).
Original entry on oeis.org
1, 0, 1, 3, 14, 80, 558, 4522, 41864, 436032, 5046680, 64251176, 892361520, 13426491520, 217555171568, 3776935252560, 69942048682112, 1376150998836224, 28669321699355520, 630448829825395840, 14593473117397510400, 354696400190943197184, 9031466708133617225984
Offset: 0
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With[{nn=30},CoefficientList[Series[1/(1-x/4 (Exp[2x]-1)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 02 2022 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/4*(exp(2*x)-1))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*2^(j-3)*binomial(i, j)*v[i-j+1])); v;
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a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*k!*stirling(n-k, k, 2)/(n-k)!);
A354314
Expansion of e.g.f. 1/(1 - x/3 * (exp(3 * x) - 1)).
Original entry on oeis.org
1, 0, 2, 9, 60, 495, 4986, 58401, 780984, 11749779, 196446870, 3612882933, 72484364052, 1575418827879, 36875093680530, 924769734574185, 24737895033896304, 703105981990977915, 21159355356941587470, 672148402091190649629, 22475238194908656800460
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/3*(exp(3*x)-1))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i, j)*v[i-j+1])); v;
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a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);
Showing 1-3 of 3 results.