A381365 G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + 2*x)^(2*n-1) * (x^n + 2*A(x))^(2*n-1).

1, 6, 267, 13686, 850848, 58650900, 4328042982, 334965057171, 26856046274793, 2212709064827217, 186314651055503493, 15969595037968661298, 1389302975474149478955, 122403732968608536815772, 10902910239945431586012588, 980514346017575408715296385

Offset

0,2

Comments

Conjecture: for n > 0, a(n) == 6 (mod 9).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) 1/3 = Sum_{n=-oo..+oo} (x*A(x))^n * (A(x)^n + 2*x)^(2*n-1) * (x^n + 2*A(x))^(2*n-1).

(2) 1/3 = Sum_{n=-oo..+oo} (x*A(x))^(2*n^2) / ( (1 + 2*x*A(x)^n)^(2*n+1) * (1 + 2*x^n*A(x))^(2*n+1) ).

Example

G.f.: A(x) = 1 + 6*x + 267*x^2 + 13686*x^3 + 850848*x^4 + 58650900*x^5 + 4328042982*x^6 + 334965057171*x^7 + 26856046274793*x^8 + ...

Prog

(PARI) {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef(-3/2 + 9/2*sum(m=-#V, #V, x^m*A^m * (A^m + 2*x)^(2*m-1) * (x^m + 2*A)^(2*m-1) ), #V-1); ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A380066, A381362, A381363, A381364.

Sequence in context: A244493 A221900 A229773 * A193983 A348701 A233234

Adjacent sequences: A381362 A381363 A381364 * A381366 A381367 A381369

Keyword

nonn,new

Author

Paul D. Hanna, Feb 21 2025

Status

approved