A344099 -id:A344099 - cp's OEIS frontend

A344100 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+4,5).

1, 1, 6, 27, 92, 323, 1070, 3527, 11314, 35708, 110478, 336629, 1011097, 2997233, 8778761, 25424358, 72867447, 206804742, 581573340, 1621407554, 4483701126, 12303384015, 33514076529, 90656680725, 243603875523, 650444927010, 1726229294595, 4554686670838, 11950683658941

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A000389, A000417, A028377, A258343, A305206, A344099, A344101.

nmax = 28; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 4, 5], {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, 5], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^6) ).

A344101 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+5,6).

Original entry on oeis.org

1, 1, 7, 35, 133, 511, 1869, 6797, 24095, 83938, 286734, 964348, 3196984, 10460310, 33813984, 108076908, 341821250, 1070484009, 3321584021, 10217036263, 31169524988, 94351439060, 283498600776, 845848778722, 2506779443603, 7381617323598, 21603241378334, 62853440151768

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A000428, A000579, A028377, A258343, A305206, A344099, A344100.

nmax = 27; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 5, 6], {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 + 5, 6], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 27}]

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^7) ).

cp's OEIS Frontend

A344100 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+4,5).

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