A326431 - OEIS

%I #10 Jul 11 2019 20:08:13

%S 1,3,26,393,8806,268011,10496566,509611213,29841622422,2063796756103,

%T 165781539363706,15259755609383885,1591551797382262450,

%U 186311156677551137459,24281772775240615369662,3498626608233846654660989,553893001173840022047130286,95833008154703833096894957199,18033356856862268626280345672162

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

%C More generally, the following sums are equal:

%C (1) exp(-r*(p+1)) * Sum_{n>=0} (q^n + p)^n * r^n / n!,

%C (2) exp(-r*(p+1)) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,

%C here, q = 1+x, p = 1, r = 1.

%H Paul D. Hanna, <a href="/A326431/b326431.txt">Table of n, a(n) for n = 0..300</a>

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

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

%e E.g.f.: A(x) = 1 + 3*x + 26*x^2/2! + 393*x^3/3! + 8806*x^4/4! + 268011*x^5/5! + 10496566*x^6/6! + 509611213*x^7/7! + 29841622422*x^8/8! + 2063796756103*x^9/9! + 165781539363706*x^10/10! + ...

%e such that

%e A(x) = exp(-2) * (1 + ((1+x) + 1) + ((1+x)^2 + 1)^2/2! + ((1+x)^3 + 1)^3/3! + ((1+x)^4 + 1)^4/4! + ((1+x)^5 + 1)^5/5! + ((1+x)^6 + 1)^6/6! + ...)

%e also,

%e A(x) = exp(-2) * (exp(1) + (1+x)*exp(1+x) + (1+x)^4*exp((1+x)^2)/2! + (1+x)^9*exp((1+x)^3)/3! + (1+x)^16*exp((1+x)^4)/4! + (1+x)^25*exp((1+x)^5)/5! + (1+x)^36*exp((1+x)^6)/6! + ...).

%o (PARI) /* Requires appropriate precision */

%o \p200

%o {a(n) = my(A = exp(-2) * sum(m=0,n+300, ((1+x)^m + 1 +x*O(x^n))^m / m! )); round(n!*polcoeff(A,n))}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A326432.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 09 2019