A054514 -id:A054514 - cp's OEIS frontend

A366112 Expansion of (1/x) * Series_Reversion( x*(1-x-x^5)/(1-x) ).

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 7, 14, 22, 31, 41, 103, 235, 457, 791, 1261, 2399, 5015, 10257, 19676, 35296, 65170, 127520, 256187, 507601, 969495, 1834433, 3534477, 6962249, 13809538, 27061252, 52439361, 101701035, 199152071, 393332277, 776589611, 1525416837

Offset: 0

Seiichi Manyama, Sep 29 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A046736, A054514, A215342.

Cf. A017899.

CoefficientList[InverseSeries[Series[x*(1-x-x^5)/(1-x),{x,0,41}]]/x,x] (* Stefano Spezia, Aug 21 2025 *)

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

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

A369489 Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^3)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 26, 98, 387, 1589, 6688, 28676, 124880, 550926, 2456831, 11056693, 50152457, 229050621, 1052393802, 4861062466, 22559964766, 105144660498, 491922058878, 2309456782464, 10876596029574, 51372213424194, 243283513468707, 1154929327702775, 5495105429597720

Offset: 0

Seiichi Manyama, Jan 24 2024

nonn

Index entries for reversions of series

Cf. A368962, A368966, A368968, A369011.

Cf. A054514, A369486.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1-x)*(1-x-x^3)^2)/x)

a(n, s=3, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t-u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

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

A370625 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^3) )^n.

Original entry on oeis.org

1, 0, 0, 3, 4, 5, 27, 63, 116, 354, 945, 2123, 5563, 14846, 36519, 93083, 244068, 622013, 1590318, 4131265, 10658969, 27440808, 71127683, 184324461, 476969939, 1237420755, 3213687698, 8343223779, 21682184311, 56400917786, 146742491187, 381991981659

Offset: 0

Seiichi Manyama, May 01 2024

nonn

Cf. A054514.

a(n, s=3, t=1, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x-x^3) / (1-x) ).

cp's OEIS Frontend

A366112 Expansion of (1/x) * Series_Reversion( x*(1-x-x^5)/(1-x) ).

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A369489 Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^3)^2 ).

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A370625 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^3) )^n.

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