A192132 -id:A192132 - cp's OEIS frontend

A369126 Expansion of (1/x) * Series_Reversion( x / ((1+x)^4+x^4) ).

1, 4, 22, 140, 970, 7104, 54096, 424008, 3398224, 27721024, 229410328, 1921308272, 16253502512, 138683973120, 1192142838656, 10314377770720, 89749921081280, 784913791336192, 6895599255571840, 60825440855493376, 538507243041624864, 4783482648574893056

Offset: 0

Seiichi Manyama, Jan 13 2024

nonn

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

Cf. A071356, A192132, A369128.

Cf. A127902.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^4+x^4))/x)

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

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

D-finite with recurrence -3*(3*n+2)*(3*n+4)*(1746*n-6043)*(n+1)*a(n) +4*(519282*n^4 -1632448*n^3 +319539*n^2 +77803*n-72516)*a(n-1) +16*(-1055610*n^4 +5245655*n^3 -8423433*n^2 +5306215*n-1129842)*a(n-2) +96*(n-2) *(150552*n^3 -673240*n^2 +868987*n -301954)*a(n-3) -64*(n-2) *(n-3) *(174726*n^2 -528221*n +220460)*a(n-4) -512*(7353*n-3733)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 24 2024

A369128 Expansion of (1/x) * Series_Reversion( x / ((1+x)^5+x^5) ).

Original entry on oeis.org

1, 5, 35, 285, 2530, 23752, 231910, 2331040, 23960235, 250692365, 2661086895, 28587333725, 310217791590, 3395464391870, 37442295427120, 415570885425280, 4638842010800025, 52044582325415025, 586553425250933055, 6637525235622842585, 75387741117556006435

Offset: 0

Seiichi Manyama, Jan 14 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..924
Index entries for reversions of series

Cf. A071356, A192132, A369126.

Cf. A364522.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^5+x^5))/x)

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

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

A369212 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^3) ).

Original entry on oeis.org

1, 2, 5, 15, 50, 177, 652, 2473, 9594, 37892, 151846, 615859, 2523217, 10427471, 43415259, 181941198, 766841846, 3248517320, 13823977350, 59067577266, 253315964424, 1089998388418, 4704475230340, 20361365646315, 88351705071583, 384280788724692, 1675063399090659

Offset: 0

Seiichi Manyama, Jan 16 2024

nonn

Index entries for reversions of series

Cf. A002212, A071356, A369158.

Cf. A071879, A192132.

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

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

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

A370287 Coefficient of x^n in the expansion of ( (1+x)^3 + x^3 )^n.

Original entry on oeis.org

1, 3, 15, 87, 531, 3333, 21309, 138015, 902547, 5946153, 39406005, 262404585, 1754316045, 11767931451, 79165530375, 533883963567, 3608242091091, 24432635451465, 165721028062605, 1125743155558677, 7657535953619721, 52151890274636463, 355576809975214095

Offset: 0

Seiichi Manyama, Feb 14 2024

nonn

Cf. A116411, A370285.
Cf. A188686, A370286.
Cf. A192132.

a(n) = sum(k=0, n\3, binomial(n, k)*binomial(3*n-3*k, n-3*k));

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

cp's OEIS Frontend

A369126 Expansion of (1/x) * Series_Reversion( x / ((1+x)^4+x^4) ).

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A369128 Expansion of (1/x) * Series_Reversion( x / ((1+x)^5+x^5) ).

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A369212 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^3) ).

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A370287 Coefficient of x^n in the expansion of ( (1+x)^3 + x^3 )^n.

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