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A181082 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n+k) * x^k] * x^n/n ).

%I A181082 #5 Apr 05 2021 21:09:40
%S A181082 1,1,2,6,34,375,15200,2066401,450054919,199271253643,431399012916702,
%T A181082 2151987403947457136,15451465958263071713102,
%U A181082 331187643758039140349444047,33475597220485400781283541412048
%N A181082 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n+k) * x^k] * x^n/n ).
%C A181082 Conjecture: this sequence consists entirely of integers.
%H A181082 G. C. Greubel, <a href="/A181082/b181082.txt">Table of n, a(n) for n = 0..70</a>
%e A181082 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 34*x^4 + 375*x^5 + 15200*x^6 +...
%e A181082 The logarithm of g.f. A(x) begins:
%e A181082   log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 111*x^4/4 + 1686*x^5/5 + 88737*x^6/6 + ... + A181083(n)*x^n/n + ...
%e A181082 and equals the series:
%e A181082   log(A(x)) = (1 + x)*x + (1 + 2^3*x + x^2)*x^2/2
%e A181082   + (1 + 3^4*x +  3^5*x^2 +      x^3)*x^3/3
%e A181082   + (1 + 4^5*x +  6^6*x^2 +  4^7*x^3 +       x^4)*x^4/4
%e A181082   + (1 + 5^6*x + 10^7*x^2 + 10^8*x^3 +   5^9*x^4 +      x^5)*x^5/5
%e A181082   + (1 + 6^7*x + 15^8*x^2 + 20^9*x^3 + 15^10*x^4 + 6^11*x^5 + x^6)*x^6/6 + ...
%t A181082 With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(n+k)*x^(n+k)/n, {k, 0,n}], {n,m+1}]], {x,0,m}], x]] (* _G. C. Greubel_, Apr 05 2021 *)
%o A181082 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k)*x^k)*x^m/m)+x*O(x^n)), n)}
%o A181082 (Magma)
%o A181082 m:=20;
%o A181082 R<x>:=PowerSeriesRing(Integers(), m);
%o A181082 Coefficients(R!( Exp( (&+[ (&+[ Binomial(n,k)^(n+k)*x^(n+k)/n : k in [0..n]]): n in [1..m+1]]) ) )); // _G. C. Greubel_, Apr 05 2021
%o A181082 (Sage)
%o A181082 m=20;
%o A181082 def A181082_list(prec):
%o A181082     P.<x> = PowerSeriesRing(ZZ, prec)
%o A181082     return P( exp( sum( sum( binomial(n,k)^(n+k)*x^(n+k)/n for k in (0..n) ) for n in (1..m+1)) ) ).list()
%o A181082 A181082_list(m) # _G. C. Greubel_, Apr 05 2021
%Y A181082 Cf. A181083 (log), variants: A181080, A181084.
%K A181082 nonn
%O A181082 0,3
%A A181082 _Paul D. Hanna_, Oct 02 2010