A181080 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k+1) * x^k] * x^n/n ).

1, 1, 2, 4, 14, 83, 774, 10641, 255918, 14643874, 1752083557, 320079087261, 79294841767020, 27407454296637142, 16895839815165609994, 26064121763003372842186, 82824096391548076720149081

Offset: 0

Author

Paul D. Hanna, Oct 02 2010

Keywords

• nonn

Comments

Conjecture: this sequence consists entirely of integers.

Note that the following g.f. does NOT yield an integer series:

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

Examples

			G.f. A(x) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 +...
The logarithm of g.f. A(x) begins:
  log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 336*x^5/5 + 4077*x^6/6 + ... + A181081(n)*x^n/n + ...
and equals the series:
  log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
  + (1 + 3^3*x +  3^2*x^2 +      x^3)*x^3/3
  + (1 + 4^4*x +  6^3*x^2 +  4^2*x^3 +      x^4)*x^4/4
  + (1 + 5^5*x + 10^4*x^2 + 10^3*x^3 +  5^2*x^4 +     x^5)*x^5/5
  + (1 + 6^6*x + 15^5*x^2 + 20^4*x^3 + 15^3*x^4 + 6^2*x^5 + x^6)*x^6/6 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..90

Crossrefs

Variants: A166894, A181070, A181082.

Cf. A181081 (log).

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 A181066_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()
A181066_list(m) # G. C. Greubel, Apr 05 2021