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A181080 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k+1) * x^k] * x^n/n ).

%I A181080 #5 Apr 05 2021 20:57:19
%S A181080 1,1,2,4,14,83,774,10641,255918,14643874,1752083557,320079087261,
%T A181080 79294841767020,27407454296637142,16895839815165609994,
%U A181080 26064121763003372842186,82824096391548076720149081
%N A181080 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k+1) * x^k] * x^n/n ).
%C A181080 Conjecture: this sequence consists entirely of integers.
%C A181080 Note that the following g.f. does NOT yield an integer series:
%C A181080   exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k) * x^k] * x^n/n ).
%H A181080 G. C. Greubel, <a href="/A181080/b181080.txt">Table of n, a(n) for n = 0..90</a>
%e A181080 G.f. A(x) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 +...
%e A181080 The logarithm of g.f. A(x) begins:
%e A181080   log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 336*x^5/5 + 4077*x^6/6 + ... + A181081(n)*x^n/n + ...
%e A181080 and equals the series:
%e A181080   log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
%e A181080   + (1 + 3^3*x +  3^2*x^2 +      x^3)*x^3/3
%e A181080   + (1 + 4^4*x +  6^3*x^2 +  4^2*x^3 +      x^4)*x^4/4
%e A181080   + (1 + 5^5*x + 10^4*x^2 + 10^3*x^3 +  5^2*x^4 +     x^5)*x^5/5
%e A181080   + (1 + 6^6*x + 15^5*x^2 + 20^4*x^3 + 15^3*x^4 + 6^2*x^5 + x^6)*x^6/6 + ...
%t A181080 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 A181080 (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 A181080 (Magma)
%o A181080 m:=20;
%o A181080 R<x>:=PowerSeriesRing(Integers(), m);
%o A181080 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 A181080 (Sage)
%o A181080 m=20;
%o A181080 def A181066_list(prec):
%o A181080     P.<x> = PowerSeriesRing(ZZ, prec)
%o A181080     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 A181080 A181066_list(m) # _G. C. Greubel_, Apr 05 2021
%Y A181080 Variants: A166894, A181070, A181082.
%Y A181080 Cf. A181081 (log).
%K A181080 nonn
%O A181080 0,3
%A A181080 _Paul D. Hanna_, Oct 02 2010