A294503 -id:A294503 - cp's OEIS frontend

A294502 Binomial transform of A026007.

1, 2, 5, 15, 45, 132, 381, 1086, 3060, 8531, 23563, 64560, 175639, 474790, 1275929, 3410180, 9068075, 23998671, 63230680, 165904474, 433596795, 1129037237, 2929620046, 7576584801, 19532878559, 50205938903, 128676829149, 328895341731, 838453003422

Offset: 0

Vaclav Kotesovec, Nov 01 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2960

Cf. A026007, A218481, A294500, A294503, A294504.

nmax = 40; s = CoefficientList[Series[Product[(1+x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

a(n) = Sum_{k=0..n} binomial(n,k) * A026007(k).
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 4 + (3*Zeta(3))^(2/3) * n^(1/3) / 8 - Zeta(3)/16) * Zeta(3)^(1/6) * 2^(n - 1/12) / (3^(1/3) * sqrt(Pi) * n^(2/3)).
G.f.: (1/(1 - x))*Product_{k>=1} (1 + x^k/(1 - x)^k)^k. - Ilya Gutkovskiy, Aug 19 2018

A294501 Inverse binomial transform of the number of planar partitions (A000219).

Original entry on oeis.org

1, 0, 2, -1, 4, -7, 19, -48, 123, -304, 728, -1694, 3865, -8735, 19739, -44875, 102818, -236939, 546988, -1260023, 2888607, -6584008, 14927816, -33714166, 75976024, -171095098, 385405617, -868708176, 1959010348, -4417777937, 9957188242, -22420045445

Offset: 0

Vaclav Kotesovec, Nov 01 2017

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A281425, A293467, A294500, A294503, A294505.

nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[(-1)^(n-k) * Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000219(k).

G.f.: (1/(1 + x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 + x)^k)). - Ilya Gutkovskiy, Aug 20 2018

A294505 Inverse binomial transform of A156616.

Original entry on oeis.org

1, 1, 3, 3, 3, 7, -3, 13, -5, -7, 49, -97, 93, 155, -997, 2893, -5989, 9007, -7121, -10805, 63305, -169375, 321137, -418503, 152653, 1142657, -4565939, 11378145, -21893565, 32887315, -33140953, -1985517, 113177979, -348817177, 734074637, -1210600023

Offset: 0

Vaclav Kotesovec, Nov 01 2017

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A156616, A294501, A294503, A294504.

nmax = 40; s = CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[(-1)^(n-k) * Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A156616(k).

G.f.: (1/(1 + x))*exp(Sum_{k>=1} (sigma_2(2*k) - sigma_2(k))*x^k/(2*k*(1 + x)^k)). - Ilya Gutkovskiy, Oct 15 2018

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

Original entry on oeis.org

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).

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A294502 Binomial transform of A026007.

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A294501 Inverse binomial transform of the number of planar partitions (A000219).

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A294505 Inverse binomial transform of A156616.

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

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