A213405 G.f.: exp( Sum_{n>=1} binomial(10*n-1, 5*n) * x^n/n ).

1, 126, 54127, 32006130, 21932146139, 16361554045542, 12899454646949132, 10572670991255846304, 8918668730118452570305, 7692248193351420093481862, 6752486830867475508568486796, 6013184272780892846457637247036, 5418931042748331247016688462113387

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..330

Feihu Liu and Guoce Xin, Simple Generating Functions for Certain Young Tableaux with Periodic Walls, arXiv:2401.14627 [math.CO], 2024.

Formula

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).

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

Example

G.f.: A(x) = 1 + 126*x + 54127*x^2 + 32006130*x^3 + 21932146139*x^4 +...
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
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(10*m-1, 5*m)*x^m/m)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

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

Sequence in context: A165028 A364402 A295816 * A178189 A365026 A294852

Adjacent sequences: A213402 A213403 A213404 * A213406 A213407 A213408

Keyword

nonn

Author

Paul D. Hanna, Jun 10 2012

Status

approved