A378733 G.f. A(x) satisfies A(x) = 1 + x / (1 - x*A(x)^2)^4.

1, 1, 4, 18, 96, 551, 3332, 20906, 134820, 888151, 5951096, 40432550, 277892604, 1928668910, 13497833600, 95150192558, 674993798716, 4815149310441, 34519885929860, 248571425473698, 1797058507267104, 13038781500215352, 94914559729835580, 692987915940266152

Offset: 0

Author

Seiichi Manyama, Dec 06 2024

Keywords

• nonn

Crossrefs

Cf. A001764, A161634, A367236.

Cf. A365123.

Programs

Maple

A378733 := proc(n)
    add(binomial(2*(n-k)+1, k)*binomial(n+3*k-1, n-k)/(2*(n-k)+1),k=0..n) ;
end proc:
seq(A378733(n),n=0..80) ; # R. J. Mathar, Dec 15 2024

PARI

a(n, r=1, s=4, t=0, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A365123.

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).