A199816 - OEIS

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A199816

G.f.: exp( Sum_{n>=1} A000984(n)*A000172(n)/4 * x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

1, 1, 8, 101, 1639, 30665, 630225, 13836981, 319062453, 7640441894, 188534274850, 4767113222750, 122998902095908, 3228067183537455, 85960229675478804, 2317956019913480326, 63193008693741620771, 1739473925024629613227, 48292271242981605779173

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OFFSET

0,3

COMMENTS

Sum_{k=0..n} C(n,k)^2 = A000984(n) defines central binomial coefficients.

Sum_{k=0..n} C(n,k)^3 = A000172(n) defines Franel numbers.

Compare to the g.f. of the Catalan numbers (A000108): exp(Sum_{n>=1} A000984(n)/2*x^n/n) and to the g.f. of A166991: exp(Sum_{n>=1} A000172(n)/2*x^n/n).

LINKS

Table of n, a(n) for n=0..18.

FORMULA

Convolution 4th power yields A199813.

EXAMPLE

G.f.: A(x) = 1 + x + 8*x^2 + 101*x^3 + 1639*x^4 + 30665*x^5 +...

where

log(A(x)) = 1*1*x + 3*5*x^2/2 + 10*28*x^3/3 + 35*173*x^4/4 + 126*1126*x^5/5 + 462*7592*x^6/6 +...+ A000984(n)/2*A000172(n)/2*x^n/n +...

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m, m)/2*sum(k=0, m, binomial(m, k)^3)/2*x^m/m)+x*O(x^n)), n)}

CROSSREFS

Cf. A199813, A166991, A000108, A181418, A000984, A000172.