A219555 -id:A219555 - cp's OEIS frontend

A344097 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+4,4).

1, 5, 25, 120, 505, 2027, 7740, 28345, 100355, 344815, 1154130, 3773955, 12085125, 37971645, 117258755, 356386016, 1067364240, 3153415695, 9198749905, 26516197720, 75586609016, 213212467695, 595482274750, 1647568369230, 4517987288720, 12284672226583, 33133931688645

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A000332, A174002, A219555, A255052, A338645, A343200, A344098.

nmax = 26; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 4, 4], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 4, 4], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * A174002(d) ) * a(n-k).

A344098 a(n) = [x^n] Product_{k>=1} (1 + x^k)^binomial(k+n-1,n-1).

Original entry on oeis.org

1, 1, 4, 29, 221, 2027, 21022, 242209, 3060262, 41936745, 618154670, 9735013136, 162892047930, 2882449728121, 53727527279464, 1051276401060921, 21529017626095851, 460231878244308738, 10246160509840187387, 237067632496414877363, 5689786581042000827057, 141415234722601777758232

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A075197, A209668, A219555, A305206, A338645, A343200, A344097.

Table[SeriesCoefficient[Product[(1 + x^k)^Binomial[k + n - 1, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 21}]
A[n_, k_] := A[n, k] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(j/d + 1) d Binomial[d + k - 1, k - 1], {d, Divisors[j]}] A[n - j, k], {j, 1, n}]]; a[n_] := A[n, n]; Table[a[n], {n, 0, 21}]

A305628 Expansion of Product_{k>=1} 1/(1 + x^k)^(k+1).

Original entry on oeis.org

1, -2, 0, -2, 5, -2, 7, -6, 11, -20, 13, -32, 31, -50, 60, -70, 124, -112, 192, -198, 295, -364, 422, -616, 661, -1002, 1034, -1500, 1737, -2208, 2808, -3234, 4462, -4876, 6735, -7464, 9990, -11610, 14410, -17866, 20947, -27082, 30493, -40056, 45147, -58196, 66999, -83278, 99641

Offset: 0

Ilya Gutkovskiy, Aug 11 2018

sign

Convolution of A081362 and A255528.

Convolution inverse of A219555.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A081362, A219555, A255528.

nmax = 48; CoefficientList[Series[Product[1/(1 + x^k)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[(-1)^k x^k (2 - x^k)/(k (1 - x^k)^2), {k, 1, nmax}]], {x,0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d (d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} (-1)^k*x^k*(2 - x^k)/(k*(1 - x^k)^2)).

a(n) ~ (-1)^n * exp(3 * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (3 * 2^(7/3) * Zeta(3)^(1/3)) - 1/12 - Pi^4 / (864 * Zeta(3))) * A * Zeta(3)^(5/36) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Aug 21 2018

A363600 Number of partitions of n into distinct parts where there are k^2+1 kinds of part k.

Original entry on oeis.org

1, 2, 6, 20, 52, 140, 356, 880, 2123, 5016, 11610, 26400, 59130, 130476, 284216, 611592, 1301344, 2740194, 5713930, 11806144, 24184908, 49142504, 99091244, 198360536, 394342884, 778818658, 1528531702, 2982017956, 5784365082, 11158728448, 21413292868

Offset: 0

Seiichi Manyama, Jun 10 2023

Cf. A027998, A219555, A255834, A363599.

Cf. A363602.

my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(k^2+1)))

G.f.: Product_{k>=1} (1+x^k)^(k^2+1).

a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * (d^2+1) ) * a(n-k).

Name suggested by Joerg Arndt, Jun 11 2023

cp's OEIS Frontend

A344097 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+4,4).

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A344098 a(n) = [x^n] Product_{k>=1} (1 + x^k)^binomial(k+n-1,n-1).

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A305628 Expansion of Product_{k>=1} 1/(1 + x^k)^(k+1).

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A363600 Number of partitions of n into distinct parts where there are k^2+1 kinds of part k.

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