A383119 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(4*k,k).

1, 3, 21, 147, 1093, 8343, 64869, 510891, 4062277, 32539647, 262181601, 2122581123, 17252278789, 140695104943, 1150670390541, 9433965332127, 77512716483461, 638080242074447, 5261486780929209, 43450477494413751, 359308411992366513, 2974886601163646379, 24657831769475675253

Offset: 0

Author

Ilya Gutkovskiy, Apr 17 2025

Keywords

• nonn

Comments

Inverse binomial transform of A005810.

Crossrefs

Cf. A002426, A005810, A346664, A359643, A383118.

Cf. A317133.

Programs

Mathematica

Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[4 k, k], {k, 0, n}], {n, 0, 22}]
Table[(-1)^n HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {1/3, 2/3, 1}, 256/27], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(1/x) Sum[Binomial[4 k, k] (x/(1 + x))^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]

PARI

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(4*k, k)); \\ Seiichi Manyama, Apr 17 2025

Formula

G.f.: (1/x) * Sum_{k>=0} binomial(4*k,k) * (x/(1 + x))^(k+1).

a(n) = [x^n] (1 + 3*x + 6*x^2 + 4*x^3 + x^4)^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x/((1+x)^4 - x) ). See A317133. - Seiichi Manyama, Apr 17 2025

a(n) ~ 229^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Apr 17 2025