A363015 Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 + x^n)^(2*n+1).

1, 0, 5, 13, 80, 352, 1955, 10155, 56934, 316413, 1810151, 10415443, 60776075, 357233548, 2118007035, 12637190038, 75866774437, 457815076217, 2775815358337, 16900781081347, 103294693694125, 633491925784696, 3897330320229845, 24045718580772438, 148748241343153325

Offset

0,3

Links

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

Formula

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

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

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

Example

G.f.: A(x) = 1 + 5*x^2 + 13*x^3 + 80*x^4 + 352*x^5 + 1955*x^6 + 10155*x^7 + 56934*x^8 + 316413*x^9 + 1810151*x^10 + ...
such that P + Q = 3, where
P = 2 + (x*A(x))*(1+x)^3 + (x*A(x))^2*(1+x^2)^5 + (x*A(x))^3*(1+x^3)^7 + (x*A(x))^4*(1+x^4)^9 + (x*A(x))^5*(1+x^5)^11 + (x*A(x))^6*(1+x^6)^13 + ... + (x*A(x))^n*(1+x^n)^(2*n+1) + ...
Q = (1/A(x))/(1+x) + (x^2/A(x))^2/(1+x^2)^3 + (x^4/A(x))^3/(1+x^3)^5 + (x^6/A(x))^4/(1+x^4)^7 + (x^8/A(x))^5/(1+x^5)^9 + (x^10/A(x))^6/(1+x^6)^11 + ... + (x^(2*n-2)/A(x))^n/(1+x^n)^(2*n-1) + ...
Explicitly,
P = 2 + x + 4*x^2 + 9*x^3 + 45*x^4 + 176*x^5 + 948*x^6 + 4621*x^7 + 25196*x^8 + 136099*x^9 + 763772*x^10 + 4319703*x^11 + 24865914*x^12 + ...
Q = 1 - x - 4*x^2 - 9*x^3 - 45*x^4 - 176*x^5 - 948*x^6 - 4621*x^7 - 25196*x^8 - 136099*x^9 - 763772*x^10 - 4319703*x^11 - 24865914*x^12 - ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-3 + sum(m=-#A, #A, x^m * Ser(A)^m * (1 + x^m +x*O(x^n))^(2*m+1) ), #A-1); ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A113876 A333082 A214536 * A324418 A359316 A165262

Adjacent sequences: A363012 A363013 A363014 * A363016 A363017 A363018

Keyword

nonn

Author

Paul D. Hanna, May 18 2023

Status

approved