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A326272 E.g.f.: Sum_{n>=0} ((1+x)^n - 1)^n * 2^n / n!.

%I A326272 #7 Jul 06 2019 09:24:16
%S A326272 1,2,16,264,6736,240160,11214144,657138944,46862522368,3973718103552,
%T A326272 393443889049600,44826129808396288,5806491899779117056,
%U A326272 846541984240702889984,137723354275132587802624,24818755539270666795663360,4922319631768240931906584576,1068365636390386171090826297344,252495346180630403940163162472448,64688594470052384103192832427687936,17893635413553390198442202310639616000
%N A326272 E.g.f.: Sum_{n>=0} ((1+x)^n - 1)^n * 2^n / n!.
%C A326272 More generally, the following sums are equal:
%C A326272 (1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
%C A326272 (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A326272 here, q = (1+x) and p = -1, r = 2.
%C A326272 In general, let F(x) be a formal power series in x such that F(0)=1, then
%C A326272 Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =
%C A326272 Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
%C A326272 here, F(x) = exp(x), q = 1+x, p = -1, r = 2, m = 1.
%H A326272 Paul D. Hanna, <a href="/A326272/b326272.txt">Table of n, a(n) for n = 0..300</a>
%F A326272 E.g.f. may be expressed by the following sums.
%F A326272 (1) Sum_{n>=0} ((1+x)^n - 1)^n * 2^n / n!.
%F A326272 (2) Sum_{n>=0} (1+x)^(n^2) * exp(-2*(1+x)^n) * 2^n / n!.
%e A326272 E.g.f: A(x) = 1 + 2*x + 16*x^2/2! + 264*x^3/3! + 6736*x^4/4! + 240160*x^5/5! + 11214144*x^6/6! + 657138944*x^7/7! + 46862522368*x^8/8! + 3973718103552*x^9/9! + 393443889049600*x^10/10! +...
%e A326272 such that
%e A326272 A(x) = 1 + 2*((1+x) - 1) + 2^2*((1+x)^2 - 1)^2/2! + 2^3*((1+x)^3 - 1)^3/3! + 2^4*((1+x)^4 - 1)^4/4! + 2^5*((1+x)^5 - 1)^5/5! + 2^6*((1+x)^6 - 1)^6/6! + 2^7*((1+x)^7 - 1)^7/7! + ...
%e A326272 also
%e A326272 A(x) = 1 + 2*(1+x)*exp(-2*(1+x)) + 2^2*(1+x)^4*exp(-2*(1+x)^2)/2! + 2^3*(1+x)^9*exp(-2*(1+x)^3)/3! + 2^4*(1+x)^16*exp(-2*(1+x)^4)/4! + 2^5*(1+x)^25*exp(-2*(1+x)^5)/5! + 2^6*(1+x)^36*exp(-2*(1+x)^6)/6! + 2^7*(1+x)^49*exp(-2*(1+x)^7)/7! + ...
%o A326272 (PARI) {a(n)=n!*polcoeff(sum(m=0, n, 2^m*((1+x+x*O(x^n))^m-1)^m/m!), n)}
%o A326272 for(n=0, 30, print1(a(n), ", "))
%Y A326272 Cf. A192935, A326273, A326274.
%Y A326272 Cf. A326092.
%K A326272 nonn
%O A326272 0,2
%A A326272 _Paul D. Hanna_, Jun 22 2019