A181084 - OEIS

%I #8 Apr 06 2021 04:54:51

%S 1,1,2,10,92,1367,87090,20385333,6633475836,4096297538926,

%T 14834973644512627,119919823546238898903,1273371038284317852447990,

%U 41086272137585936052959008420,6982122140549374036504235218052104

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

%C Conjecture: this sequence consists entirely of integers.

%H G. C. Greubel, <a href="/A181084/b181084.txt">Table of n, a(n) for n = 0..70</a>

%e G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...

%e The logarithm of g.f. A(x) begins:

%e   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 + ...

%e and equals the series:

%e   log(A(x)) = (1 + x)*x + (1 + 2^4*x + x^2)*x^2/2

%e   + (1 + 3^5*x +  3^6*x^2 +       x^3)*x^3/3

%e   + (1 + 4^6*x +  6^7*x^2 +   4^8*x^3 +       x^4)*x^4/4

%e   + (1 + 5^7*x + 10^8*x^2 +  10^9*x^3 +  5^10*x^4 +      x^5)*x^5/5

%e   + (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 + ...

%t 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 *)

%o (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)}

%o (Magma)

%o m:=20;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o 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

%o (Sage)

%o m=20;

%o def A181084_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     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()

%o A181084_list(m) # _G. C. Greubel_, Apr 05 2021

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 28 2010