A365765 -id:A365765 - cp's OEIS frontend

A365764 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x) ).

1, 4, 25, 188, 1563, 13840, 127972, 1221260, 11938471, 118936100, 1203155633, 12325599632, 127611357300, 1333153669632, 14035828918560, 148773617605036, 1586305110768863, 17002975960876300, 183102052226442475, 1980078493171083292, 21493846031259095539

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..940

Cf. A003169, A006318, A365765, A365766.

Cf. A365751.

CoefficientList[(1/x) *InverseSeries[Series[x*(1-x)^3/(1+x),{x,0,21}]],x] (* Stefano Spezia, May 04 2025 *)

a(n) = sum(k=0, n, binomial(n+1, k)*binomial(4*n-k+2, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(4*n-k+2,n-k) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(n+1,n-k).

A365766 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x) ).

Original entry on oeis.org

1, 6, 56, 626, 7721, 101322, 1387648, 19606874, 283711805, 4183074796, 62618441024, 949174260118, 14539621490403, 224721722650224, 3500129695446816, 54882906729334378, 865664769346769005, 13725517938819785298, 218639429113140366968

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..811

Cf. A003169, A006318, A365764, A365765.

Cf. A365753.

a(n) = sum(k=0, n, binomial(n+1, k)*binomial(6*n-k+4, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(6*n-k+4,n-k) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(n+1,n-k).

cp's OEIS Frontend

A365764 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x) ).

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