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A181074 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(k+1) *x^k ] *x^n/n ).

%I A181074 #9 Aug 03 2024 11:42:09
%S A181074 1,1,2,5,23,231,5405,322799,42761356,12597156231,9136063939651,
%T A181074 14655841196011960,51639276405198967750,449212631407010945983244,
%U A181074 8871353886432410987179493370,378793180251425841753491012596531
%N A181074 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(k+1) *x^k ] *x^n/n ).
%C A181074 Conjecture: this sequence consists entirely of integers.
%C A181074 Note that the following g.f. does NOT yield an integer series:
%C A181074   exp( Sum_{n>=1} [Sum_{k>=0} C(n+k-1,k)^k * x^k] * x^n/n ).
%H A181074 G. C. Greubel, <a href="/A181074/b181074.txt">Table of n, a(n) for n = 0..75</a>
%e A181074 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 231*x^5 + 5405*x^6 +...
%e A181074 The logarithm begins:
%e A181074 log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 71*x^4/4 + 1026*x^5/5 + 30912*x^6/6 +...+ A181075(n)*x^n/n +...
%e A181074 which equals the series:
%e A181074   log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
%e A181074   + (1 + 2^2*x +  3^3*x^2 +  4^4*x^3 +   5^5*x^4 +   6^6*x^5 + ...)*x^2/2
%e A181074   + (1 + 3^2*x +  6^3*x^2 + 10^4*x^3 +  15^5*x^4 +  21^6*x^5 + ...)*x^3/3
%e A181074   + (1 + 4^2*x + 10^3*x^2 + 20^4*x^3 +  35^5*x^4 +  56^6*x^5 + ...)*x^4/4
%e A181074   + (1 + 5^2*x + 15^3*x^2 + 35^4*x^3 +  70^5*x^4 + 126^6*x^5 + ...)*x^5/5
%e A181074   + (1 + 6^2*x + 21^3*x^2 + 56^4*x^3 + 126^5*x^4 + 252^6*x^5 + ...)*x^6/6
%e A181074   + (1 + 7^2*x + 28^3*x^2 + 84^4*x^3 + 210^5*x^4 + 462^6*x^5 + ...)*x^7/7 + ...
%t A181074 With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1,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 A181074 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^(k+1)*x^k)*x^m/m)+x*O(x^n)), n)}
%o A181074 (Magma)
%o A181074 m:=30;
%o A181074 R<x>:=PowerSeriesRing(Integers(), m);
%o A181074 Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^(k+1)*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // _G. C. Greubel_, Apr 05 2021
%o A181074 (SageMath)
%o A181074 m=30;
%o A181074 def A181066_list(prec):
%o A181074     P.<x> = PowerSeriesRing(ZZ, prec)
%o A181074     return P( exp( sum( sum( binomial(n+k-1,k)^(k+1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
%o A181074 A181066_list(m) # _G. C. Greubel_, Apr 05 2021
%Y A181074 Variants: A181070, A181076, A181078, A181080.
%Y A181074 Cf. A181075 (log).
%K A181074 nonn
%O A181074 0,3
%A A181074 _Paul D. Hanna_, Oct 02 2010