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A326092 E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n!.

%I A326092 #19 Jul 06 2019 09:25:44
%S A326092 1,3,11,63,525,5883,84519,1494783,31854489,800205075,23315862339,
%T A326092 777867156927,29384670476709,1245177345486987,58718905551858015,
%U A326092 3060140159517853887,175176443950054714161,10955959246057628397987,745058168844977314910331,54857350105041217492956735,4356213264604432880789346621
%N A326092 E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n!.
%C A326092 More generally, the following sums are equal:
%C A326092 (1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
%C A326092 (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A326092 here, q = (1+x) and p = 2, r = x.
%C A326092 In general, let F(x) be a formal power series in x such that F(0)=1, then
%C A326092 Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =
%C A326092 Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
%C A326092 here, F(x) = exp(x), q = 1+x, p = 2, r = x, m = 1.
%H A326092 Paul D. Hanna, <a href="/A326092/b326092.txt">Table of n, a(n) for n = 0..300</a>
%F A326092 E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n!,
%F A326092 E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(2*x*(1+x)^n) * x^n/n!.
%F A326092 a(n) = 0 (mod 3) for n > 2.
%e A326092 E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 63*x^3/3! + 525*x^4/4! + 5883*x^5/5! + 84519*x^6/6! + 1494783*x^7/7! + 31854489*x^8/8! + 800205075*x^9/9! + 23315862339*x^10/10! + ...
%e A326092 such that
%e A326092 A(x) = 1 + ((1+x) + 2)*x + ((1+x)^2 + 2)^2*x^2/2! + ((1+x)^3 + 2)^3*x^3/3! + ((1+x)^4 + 2)^4*x^4/4! + ((1+x)^5 + 2)^5*x^5/5! + ((1+x)^6 + 2)^6*x^6/6! + ((1+x)^7 + 2)^7*x^7/7! + ...
%e A326092 also
%e A326092 A(x) = 1 + (1+x)*exp(2*x*(1+x))*x + (1+x)^4*exp(2*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(2*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(2*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(2*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(2*x*(1+x)^6)*x^6/6! + ...
%o A326092 (PARI) /* E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n! */
%o A326092 {a(n) = my(A = sum(m=0,n, ((1+x)^m + 2 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A,n)}
%o A326092 for(n=0,25, print1(a(n),", "))
%o A326092 (PARI) /* E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(2*x*(1+x)^n) * x^n/n! */
%o A326092 {a(n) = my(A = sum(m=0,n, (1+x +x*O(x^n))^(m^2) * exp(2*x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
%o A326092 for(n=0,25, print1(a(n),", "))
%Y A326092 Cf. A326096, A326093, A326094.
%Y A326092 Cf. A326272.
%K A326092 nonn
%O A326092 0,2
%A A326092 _Paul D. Hanna_, Jun 21 2019