A143550 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4*A(-x)^2.
1, 1, 2, 11, 38, 257, 1040, 7646, 33374, 256718, 1171454, 9270560, 43558064, 351490167, 1686018600, 13799914556, 67223728270, 556203232266, 2741975026412, 22880729474777, 113875773363274, 956800135969601
Offset: 0
Keywords
Examples
G.f. A(x) = 1 + x + 2*x^2 + 11*x^3 + 38*x^4 + 257*x^5 + 1040*x^6 +... Related expansions: A(x)^4 = 1 + 4*x + 14*x^2 + 72*x^3 + 333*x^4 + 1936*x^5 + 9966*x^6 +... A(-x)^2 = 1 - 2*x + 5*x^2 - 26*x^3 + 102*x^4 - 634*x^5 + 2867*x^6 -+... A(x)^2*A(-x) = 1 + x + 5*x^2 + 14*x^3 + 102*x^4 + 348*x^5 + 2867*x^6 +... A(x)*A(-x) = 1 + 3*x^2 + 58*x^4 + 1597*x^6 + 51406*x^8 + 1807747*x^10 +... [A(x)*A(-x)]^6 = 1 + 18*x^2 + 483*x^4 + 15342*x^6 + 535161*x^8 +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..370
- Vaclav Kotesovec, Recurrence
Programs
-
PARI
{a(n)=local(A=1+x*O(x^n));for(i=0,2*n,A=1+x*A^4*subst(A^2,x,-x));polcoeff(A,n)}
Formula
G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^6.
G.f. satisfies: 1 - 4*y + 6*y^2 - 4*y^3 + y^4 - 2*x*y^6 + 4*x*y^7 - x*y^8 - x*y^9 + x^2*y^12 = 0, where y=A(x). - Vaclav Kotesovec, Mar 25 2014
a(n) ~ c / (sqrt(Pi)*n^(3/2)*r^n), where r = sqrt(22444621 + 5142958*sqrt(19))/46656 = 0.143559867369277217..., c = sqrt((13 - 49/sqrt(19))/3)/3 = 0.255214437... if n is even, and c = sqrt((73 - 1/sqrt(19))/3)/15 = 0.328341701... if n is odd. - Vaclav Kotesovec, Mar 25 2014
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} (-1)^(x_1+x_2) * Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 08 2025