A213406 G.f.: exp( Sum_{n>=1} binomial(12*n-1, 6*n) * x^n/n ).

1, 462, 782761, 1841287756, 5032296741620, 14989560797138774, 47213445715209298574, 154652100584276167220568, 521484200609508028036469644, 1798155951370712836530932544856, 6311247529513572335576033066558569, 22473253520120296968203645006140445948

Offset: 0

Author

Paul D. Hanna, Jun 10 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 462*x + 782761*x^2 + 1841287756*x^3 + 5032296741620*x^4 +...
such that A(x^6) = C(x)*C(u*x)*C(u^2*x)*C(-x)*C(-u*x)*C(-u^2*x) where u = exp(2*Pi*I/6) 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^6) = G(x^3)*G(-x^3) where G(x) is the g.f. of A213403:
G(x) = 1 + 10*x + 281*x^2 + 10580*x^3 + 457700*x^4 + 21475122*x^5 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..275
• 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, A213404, A213405.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1,n,binomial(12*m-1,6*m)*x^m/m)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) satisfies: A(x^6) = C(x)*C(u*x)*C(u^2*x)*C(u^3*x)*C(u^4*x)*C(u^5*x) where u = exp(2*Pi*I/6) and C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

a(n) ~ (2-sqrt(2)) * (2-sqrt(2)*3^(1/4)) * (sqrt(3)-1) * 8^(4*n+1) / (n^(3/2)*sqrt(3*Pi)). - Vaclav Kotesovec, Jul 05 2014