A365766 -id:A365766 - 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).

A365765 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x) ).

Original entry on oeis.org

1, 5, 39, 365, 3772, 41491, 476410, 5644477, 68493324, 846937140, 10633195119, 135185288475, 1736883987836, 22516798984946, 294169295918996, 3869084306851933, 51189853304834940, 680816769653570044, 9097058255214149068, 122064057533865334100

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..865

Cf. A003169, A006318, A365764, A365766.

Cf. A365752.

CoefficientList[(1/x) *InverseSeries[Series[x*(1-x)^4/(1+x),{x,0,20}]],x] (* Stefano Spezia, May 04 2025 *)

a(n) = sum(k=0, n, binomial(n+1, k)*binomial(5*n-k+3, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(5*n-k+3,n-k) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(n+1,n-k).

A365841 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^2 ).

Original entry on oeis.org

1, 7, 75, 959, 13512, 202433, 3164018, 51010415, 842090988, 14163385916, 241843189651, 4181341506009, 73054000725300, 1287786922627590, 22876030462690500, 409093644922627407, 7358978253387945404, 133067774551068558740, 2417375777620571832476

Offset: 0

Seiichi Manyama, Sep 20 2023

nonn

Cf. A032349, A365839, A365840.

Cf. A365766.

a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(2*(n+1), n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(2*(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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A365765 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x) ).

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Mathematica

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A365841 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^2 ).

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Author

Keywords

Crossrefs

Programs

PARI

Formula