A383118 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(3*k,k).

1, 2, 10, 47, 238, 1232, 6499, 34715, 187198, 1016840, 5555560, 30497150, 168073195, 929348396, 5153362231, 28646281502, 159579236014, 890644144580, 4979200476088, 27878225498030, 156298588113088, 877350590047496, 4930273302851830, 27733610884176338

Offset: 0

Author

Ilya Gutkovskiy, Apr 17 2025

Keywords

• nonn

Comments

Inverse binomial transform of A005809.

Crossrefs

Cf. A002426, A005809, A188686, A346628, A383119.

Cf. A127897.

Programs

Mathematica

Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[3 k, k], {k, 0, n}], {n, 0, 23}]
Table[(-1)^n HypergeometricPFQ[{1/3, 2/3, -n}, {1/2, 1}, 27/4], {n, 0, 23}]
nmax = 23; CoefficientList[Series[(1/x) Sum[Binomial[3 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(3*k, k)); \\ Seiichi Manyama, Apr 17 2025

Formula

G.f.: (1/x) * Sum_{k>=0} binomial(3*k,k) * (x/(1 + x))^(k+1).

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

The g.f. x * exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals Series_Reversion( x/((1+x)^3 - x) ). See A127897. - Seiichi Manyama, Apr 17 2025

a(n) ~ 23^(n + 1/2) / (3 * sqrt(Pi*n) * 2^(2*n+1)). - Vaclav Kotesovec, Apr 17 2025