A325583 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1 + 3*x)^n - A(x))^(n+1), where A(0) = 0.

1, 5, 10, 80, 568, 4220, 38692, 369602, 3829789, 42483419, 498335248, 6168187340, 80190252964, 1090909725218, 15487454931220, 228882342189464, 3513421961681770, 55912182446264327, 920864428915749175, 15671937126462121502, 275216319427229910676, 4980676147299194153192, 92778491004412178347075, 1776939414715404683846648, 34955882406696210297175882, 705630056440779526097189330

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} x^n * ((1 + 3*x)^n - A(x))^(n+1).

(2) 1 + x = Sum_{n>=0} x^n * (1 + 3*x)^(n*(n-1)) / (1 + x*(1 + 3*x)^n*A(x))^(n+1).

FORMULA FOR TERMS.

a(n) = (-1)^n (mod 3) for n >= 0.

Example

G.f.: A(x) = x + 5*x^2 + 10*x^3 + 80*x^4 + 568*x^5 + 4220*x^6 + 38692*x^7 + 369602*x^8 + 3829789*x^9 + 42483419*x^10 + 498335248*x^11 + 6168187340*x^12 + ...
such that
1 = (1 - A(x)) + x*((1+3*x) - A(x))^2 + x^2*((1+3*x)^2 - A(x))^3 + x^3*((1+3*x)^3 - A(x))^4 + x^4*((1+3*x)^4 - A(x))^5 + x^5*((1+3*x)^5 - A(x))^6 + x^6*((1+3*x)^6 - A(x))^7 + ...

Prog

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

Crossrefs

Cf. A307940, A325582, A325584, A325585.

Sequence in context: A316305 A317260 A111992 * A344194 A120598 A200983

Adjacent sequences: A325580 A325581 A325582 * A325584 A325585 A325586

Keyword

nonn

Author

Paul D. Hanna, May 11 2019

Status

approved