A181084 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} binomial(n,k)^(n+k+1) * x^k] * x^n/n ).

1, 1, 2, 10, 92, 1367, 87090, 20385333, 6633475836, 4096297538926, 14834973644512627, 119919823546238898903, 1273371038284317852447990, 41086272137585936052959008420, 6982122140549374036504235218052104

Offset: 0

Author

Paul D. Hanna, Oct 28 2010

Keywords

• nonn

Comments

Conjecture: this sequence consists entirely of integers.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...
The logarithm of g.f. A(x) begins:
  log(A(x)) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + 513657*x^6/6 + ... + A181085(n)*x^n/n + ...
and equals the series:
  log(A(x)) = (1 + x)*x + (1 + 2^4*x + x^2)*x^2/2
  + (1 + 3^5*x +  3^6*x^2 +       x^3)*x^3/3
  + (1 + 4^6*x +  6^7*x^2 +   4^8*x^3 +       x^4)*x^4/4
  + (1 + 5^7*x + 10^8*x^2 +  10^9*x^3 +  5^10*x^4 +      x^5)*x^5/5
  + (1 + 6^8*x + 15^9*x^2 + 20^10*x^3 + 15^11*x^4 + 6^12*x^5 + x^6)*x^6/6 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..70

Crossrefs

Cf. A181085 (log), variants: A181080, A181082.

Programs

Magma

m:=20;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n,k)^(n+k+1)*x^(n+k)/n : k in [0..n]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021

Mathematica

With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(n+k+1)*x^(n+k)/n, {k,0,n}], {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, m, binomial(m,k)^(m+k+1)*x^k)*x^m/m) + x*O(x^n)), n)}

Sage

m=20;
def A181084_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n,k)^(n+k+1)*x^(n+k)/n for k in (0..n) ) for n in (1..m+1)) ) ).list()
A181084_list(m) # G. C. Greubel, Apr 05 2021