A343200 -id:A343200 - cp's OEIS frontend

A338645 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+2,2).

1, 3, 9, 29, 78, 207, 526, 1284, 3054, 7084, 16071, 35748, 78167, 168195, 356754, 746772, 1544145, 3157056, 6387114, 12795366, 25397760, 49977262, 97542936, 188912466, 363196750, 693424803, 1315161528, 2478648920, 4643337213, 8648452782, 16019345259, 29515269060, 54104712129

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

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

Cf. A000217, A027480, A028377, A217093, A219555, A343200, A344097, A344098.

nmax = 32; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 2, 2], {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 + 2, 2], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 32}]

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * A027480(d) ) * a(n-k).

a(n) ~ (7/15)^(1/8) * 2^(-21/8) * n^(-5/8) * exp((2/3)*(7/15)^(1/4)*Pi * n^(3/4) + 9*sqrt(15/7)*zeta(3) * sqrt(n) / (2*Pi^2) + ((5/7)^(1/4)*Pi / (2*3^(3/4)) - 1215*(15/7)^(1/4)*zeta(3)^2 / (28*Pi^5)) * n^(1/4) + 54675*zeta(3)^3 / (98*Pi^8) - 45*zeta(3) / (28*Pi^2)). - Vaclav Kotesovec, May 12 2021

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

Original entry on oeis.org

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}]

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A338645 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+2,2).

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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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