A381909 -id:A381909 - cp's OEIS frontend

A381908 Expansion of (1/x) * Series_Reversion( x / ((1+x) * B(x)) ), where B(x) is the g.f. of A002293.

1, 2, 9, 64, 556, 5351, 54818, 585941, 6459430, 72902748, 838174008, 9781930978, 115579403512, 1379879992445, 16620303073607, 201717610488447, 2464502123154530, 30286289207099652, 374115157763376043, 4642636869759251879, 57852132860181652189, 723592983110972398779

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381909, A381910.

Cf. A002293.

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

G.f. A(x) satisfies A(x) = (1 + x*A(x)) * B(x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+4*k+1,k) * binomial(n+1,n-k)/(n+4*k+1).
a(n) = hypergeom([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [2, (2+n)/3, (3+n)/3, (4+n)/3], -2^8/3^3). - Stefano Spezia, Mar 10 2025

A381910 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 * B(x)) ), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 4, 26, 222, 2243, 25243, 305217, 3878731, 51097713, 691596081, 9558970897, 134347855874, 1914131985782, 27582542400252, 401284140631911, 5886072268606617, 86951528919335670, 1292467847124221832, 19316795168721092789, 290107272994659617741, 4375905051887803660504

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381908, A381909.

Cf. A002293.

a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(3*n+3, n-k)/(n+4*k+1));

G.f. A(x) satisfies A(x) = (1 + x*A(x))^3 * B(x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+4*k+1,k) * binomial(3*n+3,n-k)/(n+4*k+1).
a(n) = binomial(3*(1 + n), n)*hypergeom([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [(2+n)/3, (3+n)/3, (4+n)/3, 4+2*n], -2^8/3^3)/(1 + n). - Stefano Spezia, Mar 10 2025

cp's OEIS Frontend

A381908 Expansion of (1/x) * Series_Reversion( x / ((1+x) * B(x)) ), where B(x) is the g.f. of A002293.

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