A344101 -id:A344101 - cp's OEIS frontend

A344099 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).

1, 1, 5, 20, 60, 190, 561, 1651, 4720, 13300, 36716, 99872, 267836, 708890, 1854255, 4796273, 12279445, 31135188, 78236006, 194921680, 481758832, 1181675902, 2877646681, 6959866116, 16723591530, 39934902812, 94795718409, 223741936855, 525206126933, 1226393510220

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A000332, A000391, A028377, A258343, A305206, A344100, A344101.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 3, 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 + 3, 4], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 29}]

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^5) ).

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

Original entry on oeis.org

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) ).

cp's OEIS Frontend

A344099 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).

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