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)!));
A346039
Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k))^(1/k!).
Original entry on oeis.org
1, -1, -1, 3, 1, 17, -119, 165, 1191, -21989, 169527, -317837, -7182779, 54452161, 292654649, -4320853051, -46883217705, 728176373539, 9943868087879, -166076498591597, -2748733072385043, 65290726021558089, 151614363753006601, -11661992771499644571
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))^(1/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)!))*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)!))*a(n-k)/(n-k)!));
Showing 1-3 of 3 results.