A369477 -id:A369477 - cp's OEIS frontend

A370159 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^2)^2 )^n.

1, 3, 19, 132, 963, 7228, 55264, 428067, 3347843, 26378079, 209065644, 1664967747, 13312423056, 106798422942, 859244421187, 6930167382832, 56015610380931, 453628706358333, 3679805451367471, 29895358350622638, 243204082036270588, 1980931117038586824

Offset: 0

Seiichi Manyama, Feb 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A027908, A370160.

Cf. A369477, A370194.

a[n_]:=SeriesCoefficient[((1+x)*(1+x+x^2)^2)^n,{x,0,n}]; Array[a,22,0] (* Stefano Spezia, Apr 30 2024 *)

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

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(3*n-k,n-2*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) * (1+x+x^2)^2) ). See A369477.

A369479 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^3) ).

Original entry on oeis.org

1, 4, 25, 185, 1503, 12958, 116410, 1077872, 10213954, 98574454, 965545161, 9574235477, 95920415338, 969467658540, 9872949735243, 101211280459929, 1043597450013094, 10816134194658976, 112617367970103163, 1177413807406659659, 12355753915291229596

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Cf. A106228, A369477.

Cf. A365128, A369480.

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

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

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

A369478 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^2)^2) ).

Original entry on oeis.org

1, 4, 24, 170, 1320, 10868, 93197, 823484, 7445184, 68545882, 640446224, 6057249180, 57878746750, 557903174040, 5418441862824, 52971933934834, 520869559359424, 5147999004530720, 51113415228327827, 509583784051748692, 5099262428810825568

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Index entries for reversions of series

Cf. A219537, A369480.
Cf. A143927, A369477.
Cf. A369441.

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

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

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

cp's OEIS Frontend

A370159 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^2)^2 )^n.

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A369479 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^3) ).

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A369478 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^2)^2) ).

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