A381363 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(3*n-1) * (x^n + A(x))^(3*n-1).

1, 2, 66, 1986, 70750, 2773026, 115646874, 5037785442, 226634621738, 10451025646474, 491480704359206, 23483727916864770, 1136978797530782274, 55661780274825593226, 2750906179870011022678, 137071833496453114753202, 6878951603092645315764426, 347398329676557462113960102, 17642408607091724771432417962

Offset

0,2

Comments

Conjecture: for n > 0, a(n) == 2 (mod 4).

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

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

Example

G.f.: A(x) = 1 + 2*x + 66*x^2 + 1986*x^3 + 70750*x^4 + 2773026*x^5 + 115646874*x^6 + 5037785442*x^7 + 226634621738*x^8 + ...

Prog

(PARI) {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( 4*sum(m=-#V, #V, x^m*A^m * (A^m + x)^(3*m-1) * (A + x^m)^(3*m-1) ), #V-1); ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A380066, A381362, A381364, A381365.

Sequence in context: A069865 A218433 A092884 * A230735 A361884 A261784

Adjacent sequences: A381350 A381351 A381362 * A381364 A381365 A381366

Keyword

nonn,new

Author

Paul D. Hanna, Feb 21 2025

Status

approved