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)!));
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)!));
A346037
Expansion of e.g.f. Product_{k>=1} B(x^k)^(1/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 3, 9, 41, 183, 1145, 6835, 52043, 398441, 3577291, 32395905, 342875813, 3603992759, 42817702673, 518311440987, 6897155535843, 93092680608025, 1376879589495555, 20561329595474713, 333009853668160757, 5480574201430489831, 96322698607644959065
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1)^(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!*(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.