A355427 -id:A355427 - cp's OEIS frontend

A355428 a(n) = n! * [x^n] 1/(1 - Sum_{k=1..n} (exp(k*x) - 1)/k).

1, 1, 11, 284, 13564, 1037479, 116171621, 17916010524, 3640962169776, 942959405612913, 303168464105203113, 118474395231479349050, 55306932183983923942940, 30397993745996492901617435, 19429788681469866219869997285

Offset: 0

Seiichi Manyama, Jul 01 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..226

Main diagonal of A355427.

Cf. A319508.

Table[n! * SeriesCoefficient[1/(1 + HarmonicNumber[n] + E^((n + 1)*x) * LerchPhi[E^x, 1, n + 1] + Log[1 - E^x]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 02 2022 *)

a(n) = n!*polcoef(1/(1-sum(k=1, n, (exp(k*x+x*O(x^n))-1)/k)), n);

a(n) ~ c * d^n * n^(2*n + 1/2), where d = 0.4573611067742364103005235654624761643997061199669064548746966610712579358... and c = 2.41592773370058066984975000807924527905758896927935098069320182397... - Vaclav Kotesovec, Jul 02 2022

A355425 Expansion of e.g.f. 1/(1 - Sum_{k=1..2} (exp(k*x) - 1)/k).

Original entry on oeis.org

1, 2, 11, 89, 959, 12917, 208781, 3937019, 84846899, 2057107337, 55416031601, 1642126375199, 53084324076839, 1859037341680157, 70112365228588421, 2833115932639555379, 122113252334984094779, 5592296493425013663377, 271169701559687033317241

Offset: 0

Seiichi Manyama, Jul 01 2022

nonn

Column k=2 of A355427.

Cf. A004700.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, 2, (exp(k*x)-1)/k))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (1+2^(j-1))*binomial(i, j)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} (1 + 2^(k-1)) * binomial(n,k) * a(n-k).

A355426 Expansion of e.g.f. 1/(1 - Sum_{k=1..3} (exp(k*x) - 1)/k).

Original entry on oeis.org

1, 3, 24, 284, 4476, 88178, 2084564, 57493334, 1812223276, 64262620538, 2531993864004, 109738634393534, 5188538157065276, 265761817180172498, 14659691726110341844, 866403731832477234134, 54619096812884242006476, 3658454458052874579886058

Offset: 0

Seiichi Manyama, Jul 01 2022

nonn

Column k=3 of A355427.

Cf. A004701.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, 3, (exp(k*x)-1)/k))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (1+2^(j-1)+3^(j-1))*binomial(i, j)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} (1 + 2^(k-1) + 3^(k-1)) * binomial(n,k) * a(n-k).

cp's OEIS Frontend

A355428 a(n) = n! * [x^n] 1/(1 - Sum_{k=1..n} (exp(k*x) - 1)/k).

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A355425 Expansion of e.g.f. 1/(1 - Sum_{k=1..2} (exp(k*x) - 1)/k).

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A355426 Expansion of e.g.f. 1/(1 - Sum_{k=1..3} (exp(k*x) - 1)/k).

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