A383965 Self-convolution square-root of A004381, where A004381(n) = binomial(8*n,n).

1, 4, 52, 804, 13412, 233548, 4180932, 76307228, 1412731844, 26443784224, 499310856828, 9494966722696, 181620437132820, 3491268491768400, 67396227598309788, 1305787014634864584, 25380012805871145604, 494684878753394992992, 9665968233663380580256, 189289570996914582016788

Offset: 0

Author

Vaclav Kotesovec, Jun 06 2025

Keywords

• nonn

Comments

In general, for m > 1, if Sum_{k=0..n} a(k)*a(n-k) = binomial(m*n,n), then a(n) ~ m^(m*n + 1/4) / (2^(1/4) * Gamma(1/4) * (m-1)^((m-1)*n + 1/4) * n^(3/4)).

Crossrefs

Cf. A004381, A007556, A208977, A384695.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      (binomial(8*n, n)-add(a(j)*a(n-j), j=1..n-1))/2)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 06 2025

Mathematica

nmax = 20; self = ConstantArray[0, nmax + 1]; self[[1]] = 1; self[[2]] = 4; Do[self[[k+1]] = (Binomial[8*k, k] - Sum[self[[j+1]]*self[[k - j + 1]], {j, 1, k-1}]) / (2*self[[1]]);, {k, 2, nmax}]; self

Formula

a(n) ~ 2^(24*n + 1/2) / (Gamma(1/4) * 7^(7*n + 1/4) * n^(3/4)).

From Seiichi Manyama, Aug 16 2025: (Start)

Sum_{k=0..n} a(k) * a(n-k) = A004381(n).

G.f.: 1/sqrt(1 - 8*x*g^7) where g = 1+x*g^8 is the g.f. of A007556.

G.f.: sqrt( g/(8-7*g) ) where g = 1+x*g^8 is the g.f. of A007556. (End)