A346055
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, 10, 47, 246, 1617, 11586, 97463, 891610, 9189623, 102024396, 1250714445, 16351489116, 232261545869, 3499469551402, 56582677946675, 964734301550142, 17509882651329087, 333381717125596692, 6710286637806825557, 141167551783524139468
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)!));
A345762
E.g.f.: Product_{k>=1} (1 - x^k)^(1/k!).
Original entry on oeis.org
1, -1, -1, 2, 0, 29, -135, 727, -1967, -6074, 94510, 1548051, -41361089, 408842095, 213929807, -41951737904, 130060640466, 10569226878107, -229371598130229, 3327344803563111, -31418096993670379, -383829978086171112, 17799865170898698140, 220582224147105677385
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1-x^k)^(1/k!))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, (exp(x^k)-1)/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-1)!)*x^k/k))))
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a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, sumdiv(k, d, 1/(d-1)!)*a(n-k)/(n-k)!));
A346058
Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k!)).
Original entry on oeis.org
1, -1, -1, 3, 4, 2, -69, -185, 596, 1482, 22051, -8341, -450570, -1503596, -23829233, 144974757, 150086353, 4859956733, 51013196234, -504522222442, 2572161050316, -58533039862692, 69278113622988, 342581575176372, -25348876024693055, 661312712021911319
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)!));
Showing 1-3 of 3 results.