A181078 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(n+k-1) *x^k ] *x^n/n ).

1, 1, 2, 5, 29, 657, 61207, 22168009, 29875987984, 155804714312491, 3016989471632014921, 229552430038667549657248, 64995077386747098368845127628, 73163996832774559516266954450479682

Offset: 0

Author

Paul D. Hanna, Oct 03 2010

Keywords

• nonn

Comments

Conjecture: this sequence consists entirely of integers.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 29*x^4 + 657*x^5 + 61207*x^6 +...
The logarithm begins:
  log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 95*x^4/4 + 3126*x^5/5 + 363132*x^6/6 + ... + A181079(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 + 2^2*x +  3^3*x^2 +  4^4*x^3 +    5^5*x^4 +    6^6*x^5 + ...)*x^2/2
  + (1 + 3^3*x +  6^4*x^2 + 10^5*x^3 +   15^6*x^4 +   21^7*x^5 + ...)*x^3/3
  + (1 + 4^4*x + 10^5*x^2 + 20^6*x^3 +   35^7*x^4 +   56^8*x^5 + ...)*x^4/4
  + (1 + 5^5*x + 15^6*x^2 + 35^7*x^3 +   70^8*x^4 +  126^9*x^5 + ...)*x^5/5
  + (1 + 6^6*x + 21^7*x^2 + 56^8*x^3 +  126^9*x^4 + 252^10*x^5 + ...)*x^6/6
  + (1 + 7^7*x + 28^8*x^2 + 84^9*x^3 + 210^10*x^4 + 462^11*x^5 + ...)*x^7/7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..60

Crossrefs

Cf. A181079 (log), variants: A181070, A181074, A181076.

Programs

Magma

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

Mathematica

With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1,k]^(n+k-1)*x^(n+k)/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)^(m+k-1)*x^k)*x^m/m)+x*O(x^n)), n)}

Sage

m=30;
def A181078_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n+k-1,k)^(n+k-1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181078_list(m) # G. C. Greubel, Apr 05 2021