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

%I A213406 #11 Feb 06 2024 12:21:54
%S A213406 1,462,782761,1841287756,5032296741620,14989560797138774,
%T A213406 47213445715209298574,154652100584276167220568,
%U A213406 521484200609508028036469644,1798155951370712836530932544856,6311247529513572335576033066558569,22473253520120296968203645006140445948
%N A213406 G.f.: exp( Sum_{n>=1} binomial(12*n-1, 6*n) * x^n/n ).
%H A213406 Vaclav Kotesovec, <a href="/A213406/b213406.txt">Table of n, a(n) for n = 0..275</a>
%H A213406 Feihu Liu and Guoce Xin, <a href="https://arxiv.org/abs/2401.14627">Simple Generating Functions for Certain Young Tableaux with Periodic Walls</a>, arXiv:2401.14627 [math.CO], 2024.
%F A213406 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).
%F A213406 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
%e A213406 G.f.: A(x) = 1 + 462*x + 782761*x^2 + 1841287756*x^3 + 5032296741620*x^4 +...
%e A213406 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
%e A213406 C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...
%e A213406 Also, A(x^6) = G(x^3)*G(-x^3) where G(x) is the g.f. of A213403:
%e A213406 G(x) = 1 + 10*x + 281*x^2 + 10580*x^3 + 457700*x^4 + 21475122*x^5 +...
%o A213406 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,binomial(12*m-1,6*m)*x^m/m)+x*O(x^n)),n)}
%o A213406 for(n=0,20,print1(a(n),", "))
%Y A213406 Cf. A000108, A079489, A213403, A213404, A213405.
%K A213406 nonn
%O A213406 0,2
%A A213406 _Paul D. Hanna_, Jun 10 2012