A213405 - OEIS

%I #11 Feb 06 2024 12:21:58

%S 1,126,54127,32006130,21932146139,16361554045542,12899454646949132,

%T 10572670991255846304,8918668730118452570305,

%U 7692248193351420093481862,6752486830867475508568486796,6013184272780892846457637247036,5418931042748331247016688462113387

%N G.f.: exp( Sum_{n>=1} binomial(10*n-1, 5*n) * x^n/n ).

%H Vaclav Kotesovec, <a href="/A213405/b213405.txt">Table of n, a(n) for n = 0..330</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^5) = C(x)*C(u*x)*C(u^2*x)*C(u^3*x)*C(u^4*x) where u = exp(2*Pi*I/5) and C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

%F a(n) ~ (1 + sqrt((5+sqrt(5))/2) - sqrt((5+sqrt(5))/2 + sqrt(2*(5+sqrt(5))))) * (1 + sqrt((5-sqrt(5))/2) - sqrt((5-sqrt(5))/2 + sqrt(2*(5-sqrt(5))))) * 2^(10*n+4) / (sqrt(5*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 05 2014

%e G.f.: A(x) = 1 + 126*x + 54127*x^2 + 32006130*x^3 + 21932146139*x^4 +...

%e such that A(x^5) = C(x)*C(u*x)*C(u^2*x)*C(u^3*x)*C(u^4*x) where u = exp(2*Pi*I/5) and

%e C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,binomial(10*m-1,5*m)*x^m/m)+x*O(x^n)),n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A000108, A079489, A213403, A213404, A213406.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 10 2012