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A181070 Expansion of G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^(k+1)*x^k)*x^n/n ).

%I A181070 #10 Apr 05 2021 20:40:13
%S A181070 1,1,2,4,8,23,88,379,3044,32116,379279,9160509,237458908,7651718328,
%T A181070 495105710770,29747390685988,2718143583980173,436044028162542425,
%U A181070 61494671526637653928,16346049663440380567782,6106008029903796482509688
%N A181070 Expansion of G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^(k+1)*x^k)*x^n/n ).
%C A181070 Conjecture: this sequence consists entirely of integers.
%C A181070 Note that the following g.f. does NOT yield an integer series:
%C A181070 exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^k * x^k) * x^n/n ).
%H A181070 G. C. Greubel, <a href="/A181070/b181070.txt">Table of n, a(n) for n = 0..120</a>
%e A181070 G.f. A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 23*x^5 + 88*x^6 + ...
%e A181070 The logarithm of g.f. A(x) begins:
%e A181070   log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 66*x^5/5 + 357*x^6/6 + 1891*x^7/7    + ... + A181071(n)*x^n/n + ...
%e A181070 and equals the series:
%e A181070   log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
%e A181070   + (1 + 3^2*x +  3^3*x^2 +      x^3)*x^3/3
%e A181070   + (1 + 4^2*x +  6^3*x^2 +  4^4*x^3 +      x^4)*x^4/4
%e A181070   + (1 + 5^2*x + 10^3*x^2 + 10^4*x^3 +  5^5*x^4 +     x^5)*x^5/5
%e A181070   + (1 + 6^2*x + 15^3*x^2 + 20^4*x^3 + 15^5*x^4 + 6^6*x^5 + x^6)*x^6/6 + ...
%t A181070 With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(k+1)*x^(n+k)/n, {k, 0, m+2}], {n, m+1}]], {x,0,m}], x]] (* _G. C. Greubel_, Apr 05 2021 *)
%o A181070 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m,k)^(k+1)*x^k)*x^m/m)+x*O(x^n)), n)}
%o A181070 (Magma)
%o A181070 m:=30;
%o A181070 R<x>:=PowerSeriesRing(Integers(), m);
%o A181070 Coefficients(R!( Exp( (&+[ (&+[ Binomial(n,k)^(k+1)*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // _G. C. Greubel_, Apr 05 2021
%o A181070 (Sage)
%o A181070 m=30;
%o A181070 def A181070_list(prec):
%o A181070     P.<x> = PowerSeriesRing(ZZ, prec)
%o A181070     return P( exp( sum( sum( binomial(n,k)^(k+1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
%o A181070 A181070_list(m) # _G. C. Greubel_, Apr 05 2021
%Y A181070 Cf. A181071(log), variants: A181072, A181074, A181080.
%K A181070 nonn
%O A181070 0,3
%A A181070 _Paul D. Hanna_, Oct 02 2010