A307259 -id:A307259 - cp's OEIS frontend

A307260 Expansion of (1/(1 + x)) * Product_{k>=1} (1 + k*x^k/(1 + x)^k).

1, 0, 1, 1, -4, 14, -35, 77, -161, 356, -873, 2267, -5787, 13850, -30361, 59934, -103754, 147968, -139049, -58998, 730972, -2430881, 6333238, -15548722, 39845197, -110775861, 325257904, -960503811, 2756222486, -7568564555, 19815541729, -49548068461, 118752506024

Offset: 0

Ilya Gutkovskiy, Apr 01 2019

sign

Inverse binomial transform of A022629.

Cf. A022629, A293467, A294503, A307258, A307259.

a:=series((1/(1+x))*mul(1+k*x^k/(1+x)^k,k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 03 2019

nmax = 32; CoefficientList[Series[1/(1 + x) Product[(1 + k x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A022629(k).

A307262 Expansion of Product_{k>=1} (1 + k*x^k/(1 - x)^k).

Original entry on oeis.org

1, 1, 3, 10, 29, 82, 231, 646, 1780, 4835, 13009, 34794, 92600, 245119, 644983, 1686869, 4387030, 11353686, 29261059, 75134965, 192261744, 490305251, 1246128051, 3156425284, 7969135647, 20057905672, 50339682075, 126002008265, 314604617989, 783668652379, 1947689149020

Offset: 0

Ilya Gutkovskiy, Apr 01 2019

nonn

First differences of the binomial transform of A022629.

Cf. A022629, A129519, A307259, A307261, A320564.

a:=series(mul(1+k*x^k/(1-x)^k,k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 03 2019

nmax = 30; CoefficientList[Series[Product[(1 + k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A307260 Expansion of (1/(1 + x)) * Product_{k>=1} (1 + k*x^k/(1 + x)^k).

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