A353993
Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 8, 63, 668, 7850, 115914, 1847286, 34031024, 682177464, 15049816200, 357564279600, 9212847784392, 252552128708568, 7395084613746816, 229412209982127480, 7524339637608261120, 259675490280634374720, 9418707076419411194304, 357606237255136232451264
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-k*x^k)^(1/(1-x))))
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a353992(n) = n!*sum(k=1, n, sumdiv(k, d, (k/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, a353992(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356406
a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d * (k/d)^d).
Original entry on oeis.org
1, 4, 16, 79, 443, 2968, 22216, 189698, 1792402, 18745036, 213452996, 2653142952, 35448861576, 509724975264, 7824794618208, 128006170541328, 2217950478978576, 40686737647774368, 785852762719168992, 15974195890305405696, 340376906088298319616
Offset: 1
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a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)^d)));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k/k))/(1-x)))
A356409
Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k!) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 5, 28, 203, 1756, 17802, 205010, 2644287, 37669096, 586855058, 9914829508, 180429770402, 3516313661706, 73029591042943, 1609531482261375, 37504691293842367, 920966310015565936, 23764054962685200642, 642681497080268685092, 18174504398294667649782
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k!)^(1/(1-x))))
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a356407(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!^d)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356407(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356577
Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k) )^x.
Original entry on oeis.org
1, 0, 2, 6, 28, 195, 1248, 11200, 97088, 1036602, 11477230, 142038996, 1883459928, 27044341896, 412487825540, 6745633845210, 116679466051968, 2137078798914128, 41252266236703320, 838320793571448408, 17846205347898263960, 398262850748807921856
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k)^x))
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a308345(n) = n!*sumdiv(n, d, 1/(d*(n/d)^d));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*a308345(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
A356597
Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k) )^exp(x).
Original entry on oeis.org
1, 1, 5, 26, 172, 1354, 12403, 127945, 1471006, 18589503, 255951308, 3808299648, 60871219649, 1039240205691, 18868377309780, 362838034712928, 7364831540699076, 157305165900364641, 3526069495916583260, 82744901973286823822, 2028396974232995349291
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k)^exp(x)))
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a354339(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)^d))/(n-k)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354339(j)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-5 of 5 results.