A370474 G.f. A(x) satisfies A(x) = 1 + x * A(x)^(3/2) * (1 + A(x)^(3/2)).
1, 2, 9, 54, 372, 2778, 21873, 178786, 1502649, 12904524, 112741664, 998871030, 8953443276, 81047485148, 739846170864, 6803054508702, 62954736555836, 585850907166084, 5479077065774682, 51470699845616004, 485456696541512442, 4595280949098247422
Offset: 0
Keywords
Programs
-
PARI
a(n) = sum(k=0, n, binomial(n, k)*binomial(3*n/2+3*k/2+1, n)/(3*n/2+3*k/2+1));
-
PARI
a(n, r=2, s=-1, t=5, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 12 2024
Formula
a(n) = Sum{k=0..n} binomial(n,k) * binomial(3*n/2+3*k/2+1,n)/(3*n/2+3*k/2+1).
From Seiichi Manyama, Dec 12 2024: (Start)
G.f. A(x) satisfies:
(1) A(x) = ( 1 + x*A(x)^(5/2)/(1 + x*A(x)^(3/2)) )^2.
(2) A(x) = 1/( 1 - x*A(x)^2/(1 + x*A(x)^(3/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A271469.
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(s*k,n-k)/(t*k+u*(n-k)+r). (End)