A305128 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^AH(k), where AH(k) is the k-th alternating harmonic number.

1, 1, 3, 14, 79, 539, 4663, 42468, 457945, 5433281, 71036231, 994289658, 15544425103, 253283689619, 4489180389835, 84521336758904, 1687130833152561, 35365641206048129, 790065486354237643, 18340253632236738022, 449655289227002010351, 11492300073384698090795, 306803167368168113022271

Offset

0,3

Comments

a(n)/n! is the Euler transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].

Links

Table of n, a(n) for n=0..22.

N. J. A. Sloane, Transforms

Formula

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A058313(k)/A058312(k)).

Mathematica

nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^Sum[(-1)^(j + 1)/j, {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d ((-1)^(d + 1) LerchPhi[-1, 1, d + 1] + Log[2]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Crossrefs

Cf. A028342, A058312, A058313, A168243, A303970.

Sequence in context: A020089 A218677 A353079 * A027614 A306040 A168592

Adjacent sequences: A305125 A305126 A305127 * A305129 A305130 A305131

Keyword

nonn

Author

Ilya Gutkovskiy, May 26 2018

Status

approved