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%I #16 Feb 06 2024 12:22:01
%S 1,35,3830,570451,98118690,18345127262,3621992085708,743083237338755,
%T 156855468465746346,33846364485841559594,7432235142547456907188,
%U 1655432795976620159935790,373110570133205997324473492,84936332285861009708851200092,19500719075082334054293510927128
%N G.f.: exp( Sum_{n>=1} binomial(8*n-1, 4*n) * x^n/n ).
%H Vaclav Kotesovec, <a href="/A213404/b213404.txt">Table of n, a(n) for n = 0..410</a>
%H 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 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).
%F a(n) ~ (1-sqrt(2*(sqrt(2)-1))) * 4^(4*n+1) / (n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Jul 05 2014
%e G.f.: A(x) = 1 + 35*x + 3830*x^2 + 570451*x^3 + 98118690*x^4 +...
%e such that A(x^4) = C(x)*C(-x)*C(I*x)*C(-I*x) where I^2 = -1 and
%e C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...
%e Also, A(x^2) = G(x)*G(-x) where G(x) is the g.f. of A079489:
%e G(x) = 1 + 3*x + 22*x^2 + 211*x^3 + 2306*x^4 + 27230*x^5 + 338444*x^6 +...
%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,binomial(8*m-1,4*m)*x^m/m)+x*O(x^n)),n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A000108, A079489, A213403, A213405, A213406.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 10 2012
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