A143550 G.f. 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

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..370

Vaclav Kotesovec, Recurrence

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

Example

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 +...

Prog

(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)}

Crossrefs

Cf. A143338, A143549, A143551, A143552, A143553, A143554.

Sequence in context: A196850 A203534 A166989 * A259213 A259658 A350952

Adjacent sequences: A143547 A143548 A143549 * A143551 A143552 A143553

Keyword

nonn

Author

Paul D. Hanna, Aug 24 2008

Status

approved