A250917 E.g.f.: exp( x*C(x)^3 ) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, A000108.

1, 1, 7, 73, 1033, 18541, 403831, 10351237, 305355793, 10192132153, 379819484551, 15634219476481, 704566985120857, 34506514429777573, 1825081888365736183, 103685565729559782781, 6297505655719537293601, 407233553972252986277617, 27935786938445348562454663

Offset

0,3

Comments

In general, if k>0 and e.g.f. = exp(x*C(x)^k) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, then a(n) ~ k * 2^(2*n + k - 5/2) * n^(n-1) / exp(n - 2^(k-2)). - Vaclav Kotesovec, Aug 22 2017

Links

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

Formula

a(n) = Sum_{k=0..n} n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k) for n>0 with a(0)=1.

a(n) ~ 3 * 2^(2*n+1/2) * n^(n-1) / exp(n-2). - Vaclav Kotesovec, Aug 22 2017

Conjecture D-finite with recurrence: +2*a(n) +(-11*n+20)*a(n-1) +(n^3+9*n^2-116*n+164)*a(n-2) +(-4*n^4+35*n^3+n^2-317*n+342)*a(n-3) -6*(n-3)*(6*n^3-50*n^2+147*n-176)*a(n-4) +12*(n-5)*(2*n-9)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 25 2020

Example

E.g.f.: A(x) = 1 + x + 7*x^2/2! + 73*x^3/3! + 1033*x^4/4! + 18541*x^5/5! +...
such that log(A(x)) = x*C(x)^3,
log(A(x)) = x + 3*x^2 + 9*x^3 + 28*x^4 + 90*x^5 + 297*x^6 + 1001*x^7 +...
where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Prog

(PARI) {a(n)=local(C=1); for(i=1, n, C=1+x*C^2 +x*O(x^n));
n!*polcoeff(exp(x*C^3), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if(n==0, 1, sum(k=0, n, n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k) ))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A251568, A000108, A001517.

Sequence in context: A321837 A050352 A261783 * A112939 A058350 A048174

Adjacent sequences: A250914 A250915 A250916 * A250918 A250919 A250920

Keyword

nonn

Author

Paul D. Hanna, Dec 06 2014

Status

approved