A181066 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^3 *x^k ] *x^n/n ).

1, 1, 2, 7, 31, 157, 865, 5051, 30774, 193669, 1250319, 8240232, 55239187, 375624781, 2585449450, 17982937876, 126222946496, 893073250063, 6363674671524, 45631735776036, 329065051395940, 2385126419825231

Offset: 0

Author

Paul D. Hanna, Oct 03 2010

Keywords

• nonn

Comments

Compare g.f. to a g.f. of the Catalan numbers (A000108):

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

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 157*x^5 + 865*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 95*x^4/4 + 606*x^5/5 + 4032*x^6/6 +...+ A181067(n)*x^n/n +...
which equals the series:
log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
+ (1 + 2^3*x +  3^3*x^2 +  4^3*x^3 +   5^3*x^4 +   6^3*x^5 + ...)*x^2/2
+ (1 + 3^3*x +  6^3*x^2 + 10^3*x^3 +  15^3*x^4 +  21^3*x^5 + ...)*x^3/3
+ (1 + 4^3*x + 10^3*x^2 + 20^3*x^3 +  35^3*x^4 +  56^3*x^5 + ...)*x^4/4
+ (1 + 5^3*x + 15^3*x^2 + 35^3*x^3 +  70^3*x^4 + 126^3*x^5 + ...)*x^5/5
+ (1 + 6^3*x + 21^3*x^2 + 56^3*x^3 + 126^3*x^4 + 252^3*x^5 + ...)*x^6/6
+ (1 + 7^3*x + 28^3*x^2 + 84^3*x^3 + 210^3*x^4 + 462^3*x^5 + ...)*x^7/7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000108, A181067 (log), A181068 (variant).

Programs

Magma

m:=30;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^3*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021

Mathematica

With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1, k]^3*x^k*x^n/n, {k, 0, m+2}], {n, m+1}]], {x,0,m}], x]] (* G. C. Greubel, Apr 05 2021 *)

PARI

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

Sage

m=30;
def A181066_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n+k-1,k)^3*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181066_list(m) # G. C. Greubel, Apr 05 2021