A213404 G.f.: exp( Sum_{n>=1} binomial(8*n-1, 4*n) * x^n/n ).

1, 35, 3830, 570451, 98118690, 18345127262, 3621992085708, 743083237338755, 156855468465746346, 33846364485841559594, 7432235142547456907188, 1655432795976620159935790, 373110570133205997324473492, 84936332285861009708851200092, 19500719075082334054293510927128

Offset: 0

Author

Paul D. Hanna, Jun 10 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 35*x + 3830*x^2 + 570451*x^3 + 98118690*x^4 +...
such that A(x^4) = C(x)*C(-x)*C(I*x)*C(-I*x) where I^2 = -1 and
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...
Also, A(x^2) = G(x)*G(-x) where G(x) is the g.f. of A079489:
G(x) = 1 + 3*x + 22*x^2 + 211*x^3 + 2306*x^4 + 27230*x^5 + 338444*x^6 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..410
• Feihu Liu and Guoce Xin, Simple Generating Functions for Certain Young Tableaux with Periodic Walls, arXiv:2401.14627 [math.CO], 2024.

Crossrefs

Cf. A000108, A079489, A213403, A213405, A213406.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1,n,binomial(8*m-1,4*m)*x^m/m)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) satisfies: A(x^4) = C(x)*C(-x)*C(I*x)*C(-I*x) where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

a(n) ~ (1-sqrt(2*(sqrt(2)-1))) * 4^(4*n+1) / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jul 05 2014