A181068 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^4 *x^k ] *x^n/n ).

1, 1, 2, 11, 80, 714, 7095, 76206, 864590, 10227727, 125001862, 1568419058, 20108619244, 262510020319, 3479914302802, 46742907726147, 635092339459857, 8716058291255777, 120686879727465365, 1684357785848110976

Offset: 0

Author

Paul D. Hanna, Oct 08 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 + 11*x^3 + 80*x^4 + 714*x^5 + 7095*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 28*x^3/3 + 275*x^4/4 + 3126*x^5/5 + 37632*x^6/6 +...+ A181069(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 + 2^4*x +  3^4*x^2 +  4^4*x^3 +   5^4*x^4 +   6^4*x^5 +...)*x^2/2
+ (1 + 3^4*x +  6^4*x^2 + 10^4*x^3 +  15^4*x^4 +  21^4*x^5 +...)*x^3/3
+ (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 +  35^4*x^4 +  56^4*x^5 +...)*x^4/4
+ (1 + 5^4*x + 15^4*x^2 + 35^4*x^3 +  70^4*x^4 + 126^4*x^5 +...)*x^5/5
+ (1 + 6^4*x + 21^4*x^2 + 56^4*x^3 + 126^4*x^4 + 252^4*x^5 +...)*x^6/6
+ (1 + 7^4*x + 28^4*x^2 + 84^4*x^3 + 210^4*x^4 + 462^4*x^5 +...)*x^7/7 + ...
		

Links

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

Crossrefs

Cf. A000108, A181069 (log), A181066 (variant).

Programs

Magma

m:=30;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^4*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]^4*x^(n+k)/n, {k,0,m+2}], {n,1,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)^4*x^k)*x^m/m)+x*O(x^n)), n)}

Sage

m=30;
def A181068_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n+k-1,k)^4*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181068_list(m) # G. C. Greubel, Apr 05 2021

Formula

a(n) ~ c * 16^n / n^(5/2), where c = 0.034183651246881715583041336040447549489320454248320978... - Vaclav Kotesovec, Apr 05 2021