A357232 a(n) = coefficient of x^n, n >= 0, in A(x) such that: 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^n)^(2*n+1).

1, 3, 25, 254, 2763, 32180, 393169, 4964017, 64254694, 848214039, 11375359344, 154547261539, 2122630191360, 29423373611509, 411105855956011, 5783709944279141, 81862107418919278, 1164873718427628846, 16654829725736560441, 239140138388082634266, 3446933945466334214525

Offset

0,2

Comments

Compare to A355865.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

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

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

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

Example

G.f.: A(x) = 1 + 3*x + 25*x^2 + 254*x^3 + 2763*x^4 + 32180*x^5 + 393169*x^6 + 4964017*x^7 + 64254694*x^8 + 848214039*x^9 + 11375359344*x^10 + ...

Prog

(PARI) {a(n) = my(A=1); for(L=1, n, A = truncate(A);
A = A + 1 - (1/2)*sum(m=-L, L, (-1)^m * x^m * (2*A + x^m +x^2*O(x^(L+1)))^(2*m+1) ) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A355865, A355868, A357402.

Sequence in context: A086655 A064326 A006091 * A355865 A009042 A080896

Adjacent sequences: A357229 A357230 A357231 * A357233 A357234 A357235

Keyword

nonn

Author

Paul D. Hanna, Oct 14 2022

Status

approved