A381364 G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + 2*x)^(n-1) * (x^n + 2*A(x))^(n-1).

1, 6, 69, 1185, 25971, 638664, 16870146, 469957290, 13643527371, 409333196409, 12617508055164, 397955799543372, 12805103784047244, 419461854027499095, 13966745624480483286, 472195682273425114437, 16198043820079506200952, 563559268046248762052514, 19883430320804741832966096

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)^(n-1) * (x^n + 2*A(x))^(n-1).

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

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)^(m-1) * (x^m + 2*A)^(m-1) ), #V-1); ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A380066, A381362, A381363, A381365.

Sequence in context: A177751 A338176 A235327 * A349556 A381479 A364982

Adjacent sequences: A381360 A381362 A381363 * A381365 A381366 A381367

Keyword

nonn

Author

Paul D. Hanna, Feb 21 2025

Status

approved