A159321 L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} (1 + C(2n-1,n-1)*x)^n * x^n/n.

1, 3, 10, 59, 676, 13782, 525624, 39289875, 5306323852, 1456575517928, 728976500267566, 736763475137343458, 1399734009767581939400, 5252418655426943548516230, 38299695673374257212534923730

Offset

1,2

Links

Table of n, a(n) for n=1..15.

Formula

a(n) = n*Sum_{k=0..[n/2]} C(n-k,k)*C(2n-2k-1,n-k-1)^k/(n-k) for n>=1.

Example

L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 59*x^4/4 + 676*x^5/5 +...
L(x) = (1+x)*x + (1+3*x)^2*x^2/2 + (1+10*x)^3*x^3/3 + (1+35*x)^4*x^4/4 +...
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 20*x^4 + 158*x^5 + 2474*x^6 +... (A159320).

Mathematica

Table[n*Sum[Binomial[n-k, k]*Binomial[2n-2k-1, n-k-1]^k/(n-k), {k, 0, Floor[n/2]}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 06 2014 *)

Prog

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

Crossrefs

Cf. A159320 (exp), A001700, A158873 (variant).

Sequence in context: A333031 A242953 A112101 * A181077 A158873 A103591

Adjacent sequences: A159318 A159319 A159320 * A159322 A159323 A159324

Keyword

nonn

Author

Paul D. Hanna, Apr 15 2009

Status

approved