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A386646 Expansion of e.g.f. Sum_{n>=0} (2^n + x)^n * x^n / n!.

%I A386646 #15 Aug 10 2025 11:55:27
%S A386646 1,2,18,536,66316,33636832,68750980216,562995064353920,
%T A386646 18446990378410477200,2417856827427983647531520,
%U A386646 1267651025241922183470966470176,2658456127743272591813667810372278272,22300745369876426654206395965130496991176384,748288839162767087393170357241926671150780067340288
%N A386646 Expansion of e.g.f. Sum_{n>=0} (2^n + x)^n * x^n / n!.
%C A386646 In general, the following sums are equal:
%C A386646 (C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
%C A386646 (C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A386646 here, q = A(x) with p = x, r = x.
%H A386646 Paul D. Hanna, <a href="/A386646/b386646.txt">Table of n, a(n) for n = 0..70</a>
%F A386646 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A386646 (1) A(x) = Sum_{n>=0} (2^n + x)^n * x^n / n!.
%F A386646 (2) A(x) = Sum_{n>=0} 2^(n^2) * exp(2^n*x^2) * x^n / n!.
%F A386646 a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Aug 09 2025
%e A386646 E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 536*x^3/3! + 66316*x^4/4! + 33636832*x^5/5! + 68750980216*x^6/6! + 562995064353920*x^7/7! + 18446990378410477200*x^8/8! + ...
%e A386646 where
%e A386646 A(x) = 1 + (2 + x)*x + (2^2 + x)^2*x^2/2! + (2^3 + x)^3*x^3/n! + (2^4 + x)^4*x^4/4! + (2^5 + x)^5*x^5/5! + ...
%e A386646 Also,
%e A386646 A(x) = exp(x^2) + 2*exp(2*x^2)*x + 2^4*exp(2^2*x^2)*x^2/2! + 2^9*exp(2^3*x^2)*x^3/3! + 2^16*exp(2^4*x^2)*x^4/4! + ...
%t A386646 nmax = 15; CoefficientList[Series[Sum[(2^k + x)^k * x^k / k!, {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Aug 09 2025 *)
%o A386646 (PARI) {a(n) = my(A = sum(m=0, n, (2^m + x)^m * x^m/m! +x*O(x^n)) ); n!*polcoef(A,n)}
%o A386646 for(n=0, 15, print1(a(n), ", "))
%Y A386646 Cf. A136516.
%K A386646 nonn
%O A386646 0,2
%A A386646 _Paul D. Hanna_, Aug 08 2025