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A181078 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(n+k-1) *x^k ] *x^n/n ).

%I A181078 #5 Apr 05 2021 20:57:09
%S A181078 1,1,2,5,29,657,61207,22168009,29875987984,155804714312491,
%T A181078 3016989471632014921,229552430038667549657248,
%U A181078 64995077386747098368845127628,73163996832774559516266954450479682
%N A181078 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(n+k-1) *x^k ] *x^n/n ).
%C A181078 Conjecture: this sequence consists entirely of integers.
%H A181078 G. C. Greubel, <a href="/A181078/b181078.txt">Table of n, a(n) for n = 0..60</a>
%e A181078 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 29*x^4 + 657*x^5 + 61207*x^6 +...
%e A181078 The logarithm begins:
%e A181078   log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 95*x^4/4 + 3126*x^5/5 + 363132*x^6/6 + ... + A181079(n)*x^n/n + ...
%e A181078 which equals the series:
%e A181078   log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...)*x
%e A181078   + (1 + 2^2*x +  3^3*x^2 +  4^4*x^3 +    5^5*x^4 +    6^6*x^5 + ...)*x^2/2
%e A181078   + (1 + 3^3*x +  6^4*x^2 + 10^5*x^3 +   15^6*x^4 +   21^7*x^5 + ...)*x^3/3
%e A181078   + (1 + 4^4*x + 10^5*x^2 + 20^6*x^3 +   35^7*x^4 +   56^8*x^5 + ...)*x^4/4
%e A181078   + (1 + 5^5*x + 15^6*x^2 + 35^7*x^3 +   70^8*x^4 +  126^9*x^5 + ...)*x^5/5
%e A181078   + (1 + 6^6*x + 21^7*x^2 + 56^8*x^3 +  126^9*x^4 + 252^10*x^5 + ...)*x^6/6
%e A181078   + (1 + 7^7*x + 28^8*x^2 + 84^9*x^3 + 210^10*x^4 + 462^11*x^5 + ...)*x^7/7 + ...
%t A181078 With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1,k]^(n+k-1)*x^(n+k)/n, {k,0,m+2}], {n,m+1}]], {x,0,m}], x]] (* _G. C. Greubel_, Apr 05 2021 *)
%o A181078 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^(m+k-1)*x^k)*x^m/m)+x*O(x^n)), n)}
%o A181078 (Magma)
%o A181078 m:=30;
%o A181078 R<x>:=PowerSeriesRing(Integers(), m);
%o A181078 Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^(n+k-1)*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // _G. C. Greubel_, Apr 05 2021
%o A181078 (Sage)
%o A181078 m=30;
%o A181078 def A181078_list(prec):
%o A181078     P.<x> = PowerSeriesRing(ZZ, prec)
%o A181078     return P( exp( sum( sum( binomial(n+k-1,k)^(n+k-1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
%o A181078 A181078_list(m) # _G. C. Greubel_, Apr 05 2021
%Y A181078 Cf. A181079 (log), variants: A181070, A181074, A181076.
%K A181078 nonn
%O A181078 0,3
%A A181078 _Paul D. Hanna_, Oct 03 2010