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A326270 E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.

%I A326270 #26 Jul 09 2019 11:51:53
%S A326270 1,2,18,314,8434,314362,15278642,928696442,68509258098,5995762219514,
%T A326270 611538502747826,71656036268121978,9532232740451770866,
%U A326270 1425414297318661354746,237588200534263288095538,43821269448954050939558522,8887255081413035850889914994,1970841722610600810208914571258,475544555000142351430865220032434,124299766720856839788225909600114042,35056463298676734373530025799446104818
%N A326270 E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.
%C A326270 More generally, the following sums are equal:
%C A326270 Sum_{n>=0} (p + q^n)^n * r^n/n! =
%C A326270 Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A326270 here, q = exp(x) with p = -1, r = 2.
%C A326270 In general, let F(x) be a formal power series in x such that F(0)=1, then
%C A326270 Sum_{n>=0} m^n * F(q^n*r)^b * log( F(q^n*r) )^n / n! =
%C A326270 Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
%C A326270 here, F(x) = exp(x), q = exp(x), p = -1, r = 2, m = 1.
%H A326270 Paul D. Hanna, <a href="/A326270/b326270.txt">Table of n, a(n) for n = 0..300</a>
%F A326270 E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.
%F A326270 E.g.f.: Sum_{n>=0} 2^n * exp(n^2*x) * exp( -2*exp(n*x) ) / n!.
%F A326270 O.g.f.: Sum_{n>=0} 2^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x).
%F A326270 a(n) = Sum_{k=0..n} 2^k * k^n * Stirling2(n,k).
%e A326270 E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 314*x^3/3! + 8434*x^4/4! + 314362*x^5/5! + 15278642*x^6/6! + 928696442*x^7/7! + 68509258098*x^8/8! + 5995762219514*x^9/9! + 611538502747826*x^10/10! + ...
%e A326270 such that
%e A326270 A(x) = 1 + 2*(exp(x) - 1) + 2^2*(exp(2*x) - 1)^2/2! + 2^3*(exp(3*x) - 1)^3/3! + 2^4*(exp(4*x) - 1)^4/4! + 2^5*(exp(5*x) - 1)^5/5! + 2^6*(exp(6*x) - 1)^6/6! + ...
%e A326270 also
%e A326270 A(x) = exp(-2) + 2*exp(x)*exp(-2*exp(x)) + 2^2*exp(4*x)*exp(-2*exp(2*x))/2! + 2^3*exp(9*x)*exp(-2*exp(3*x))/3! + 2^4*exp(16*x)*exp(-2*exp(4*x))/4! + 2^5*exp(25*x)*exp(-2*exp(5*x))/5! + 2^6*exp(36*x)*exp(-2*exp(6*x))/6! + ...
%e A326270 ORDINARY GENERATING FUNCTION.
%e A326270 O.g.f.: B(x) = 1 + 2*x + 18*x^2 + 314*x^3 + 8434*x^4 + 314362*x^5 + 15278642*x^6 + 928696442*x^7 + 68509258098*x^8 + 5995762219514*x^9 + ...
%e A326270 such that
%e A326270 B(x) = 1 + 2*x/(1-x) + 2^2*2^2*x^2/((1-2*x)*(1-4*x)) + 2^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 2^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 2^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...
%e A326270 RELATED SERIES.
%e A326270 Below we illustrate the following identity at specific values of x:
%e A326270 Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!  =  Sum_{n>=0} 2^n * exp(n^2*x) * exp( -2*exp(n*x) ) / n!.
%e A326270 (1) At x = -1, the following sums are equal
%e A326270 S1 = Sum_{n>=0} (-2)^n * (1 - exp(-n))^n / n!,
%e A326270 S1 = Sum_{n>=0} 2^n * exp(-n^2) * exp( -2*exp(-n) ) / n!,
%e A326270 where S1 = 0.51596189603321982013621912500044621350106513780391377129738...
%e A326270 (2) At x = -2, the following sums are equal
%e A326270 S2 = Sum_{n>=0} (-2)^n * (1 - exp(-2*n))^n / n!,
%e A326270 S2 = Sum_{n>=0} 2^n * exp(-2*n^2) * exp( -2*exp(-2*n) ) / n!,
%e A326270 where S2 = 0.34246794778612083304129071190905516612972983097016819355092...
%e A326270 (3) At x = -log(2), the following sums are equal
%e A326270 S3 = Sum_{n>=0} 2^(-n*(n-1)) * (2^n - 1)^n * (-1)^n / n!,
%e A326270 S3 = Sum_{n>=0} 2^(-n*(n-1)) * exp( -1/2^(n-1) ) / n!,
%e A326270 where S3 = 0.58106816860114387883649557314841837351794236167582918403231...
%t A326270 Flatten[{1, Table[Sum[2^k * k^n * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* _Vaclav Kotesovec_, Jul 09 2019 *)
%o A326270 (PARI) {a(n) = sum(k=0,n, 2^k * k^n * stirling(n,k,2) )}
%o A326270 for(n=0,30,print1(a(n),", "))
%o A326270 (PARI) /* E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n! */
%o A326270 {a(n) = n! * polcoeff(sum(m=0, n, 2^m * (exp(m*x +x*O(x^n)) - 1)^m / m!), n)}
%o A326270 for(n=0,30,print1(a(n),", "))
%o A326270 (PARI) /* O.g.f.: Sum_{n>=0} 2^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
%o A326270 {a(n) = polcoeff(sum(m=0, n, 2^m * m^m * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
%o A326270 for(n=0,30,print1(a(n),", "))
%Y A326270 Cf. A108459, A326271, A326288.
%K A326270 nonn
%O A326270 0,2
%A A326270 _Paul D. Hanna_, Jun 28 2019