A346056
Expansion of e.g.f. Product_{k>=1} B(x^k/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 3, 9, 38, 168, 915, 5225, 34228, 236622, 1805297, 14498751, 125907798, 1146476984, 11129874215, 112934907867, 1209762361679, 13499714095281, 157931096158594, 1918777335806274, 24309294470496502, 318987321135326838, 4346474397776153974
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k/k!)-1))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, exp(x^k/k!)-1))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)!^d))*x^k))))
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a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!^d))*a(n-k)/(n-k)!));
A346057
Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k)).
Original entry on oeis.org
1, -1, -1, 2, 3, 14, -55, 62, -637, 338, -3861, 335312, -4499803, 43490108, -246353731, 2189950310, -47336985225, 1224524919590, -21516426400621, 346681988108648, -4499477383730851, 69294602646065900, -1418045089870455795, 45246859024830444566
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(1-exp(x^k/k)))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, 1-exp(x^k/k)))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)^d))*x^k))))
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a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)^d))*a(n-k)/(n-k)!));
A356494
Expansion of e.g.f. Product_{k>0} B(k * x^k) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 6, 35, 327, 2892, 37943, 459895, 7330172, 116054835, 2168292295, 41072348550, 898738816957, 19782331776937, 487091519709590, 12305361661242275, 337777113607935587, 9528258228302443724, 289373132780801591323, 9016757353084706862647
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(k*x^k)-1))))
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a354843(n) = n!*sumdiv(n, d, (n/d)^d/d!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354843(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356495
Expansion of e.g.f. Product_{k>0} B((k * x)^k) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 10, 191, 7287, 424292, 37434683, 4512452023, 726390985036, 149098938941283, 38187088904721655, 11903871288193251930, 4442392007373264794677, 1953788894138983864638457, 1000334575509506861927067378, 589712001176601700420819946615
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp((k*x)^k)-1))))
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a354892(n) = n!*sumdiv(n, d, d^n/(n/d)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354892(j)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-4 of 4 results.