This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A326096 #11 Jul 09 2019 01:30:29
%S A326096 1,2,6,32,256,2712,37744,645752,13371264,327748832,9332342944,
%T A326096 304875611328,11298403070464,470279355784448,21809054992366464,
%U A326096 1118931830122060928,63115145120561606656,3892675200470654980608,261242029823318546162176,18994387868664467440590848,1490356266852194536099393536,125747158151444491631754033152
%N A326096 E.g.f.: Sum_{n>=0} ((1+x)^n + 1)^n * x^n/n!.
%C A326096 More generally, the following sums are equal:
%C A326096 (1) Sum_{n>=0} (q^n + p)^n * x^n/n!,
%C A326096 (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*x) * x^n/n!;
%C A326096 here, q = (1+x) and p = 1.
%C A326096 In general, let F(x) be a formal power series in x such that F(0)=1, then
%C A326096 Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =
%C A326096 Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
%C A326096 here, F(x) = exp(x), q = 1+x, p = 1, r = x, m = 1.
%H A326096 Paul D. Hanna, <a href="/A326096/b326096.txt">Table of n, a(n) for n = 0..300</a>
%F A326096 E.g.f.: Sum_{n>=0} ((1+x)^n + 1)^n * x^n/n!,
%F A326096 E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(x*(1+x)^n) * x^n/n!.
%e A326096 E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 32*x^3/3! + 256*x^4/4! + 2712*x^5/5! + 37744*x^6/6! + 645752*x^7/7! + 13371264*x^8/8! + 327748832*x^9/9! + 9332342944*x^10/10! + 304875611328*x^11/11! + 11298403070464*x^12/12! + ...
%e A326096 such that
%e A326096 A(x) = 1 + ((1+x) + 1)*x + ((1+x)^2 + 1)^2*x^2/2! + ((1+x)^3 + 1)^3*x^3/3! + ((1+x)^4 + 1)^4*x^4/4! + ((1+x)^5 + 1)^5*x^5/5! + ((1+x)^6 + 1)^6*x^6/6! + ((1+x)^7 + 1)^7*x^7/7! + ...
%e A326096 also
%e A326096 A(x) = 1 + (1+x)*exp(x*(1+x))*x + (1+x)^4*exp(x*(1+x)^2)*x^2/2! + (1+x)^9*exp(x*(1+x)^3)*x^3/3! + (1+x)^16*exp(x*(1+x)^4)*x^4/4! + (1+x)^25*exp(x*(1+x)^5)*x^5/5! + (1+x)^36*exp(x*(1+x)^6)*x^6/6! + ...
%e A326096 RELATED SERIES.
%e A326096 Below we illustrate the following identity at specific values of x:
%e A326096 Sum_{n>=0} ((1+x)^n + 1)^n * x^n/n! = Sum_{n>=0} (1+x)^(n^2) * exp(x*(1+x)^n) * x^n/n!.
%e A326096 (1) At x = -1/2, the following sums are equal
%e A326096 S1 = Sum_{n>=0} (-1)^n * 2^(-n*(n+1)) * (2^n + 1)^n / n!,
%e A326096 S1 = Sum_{n>=0} (-1)^n * 2^(-n*(n+1)) * exp( -1/2^(n+1) ) / n!,
%e A326096 where S1 = 0.41868678468707099609788224908427981408329845879700862624389...
%e A326096 (2) At x = -2/3, the following sums are equal
%e A326096 S2 = Sum_{n>=0} (-1)^n * 2^n * 3^(-n*(n+1)) * (3^n + 1)^n / n!,
%e A326096 S2 = Sum_{n>=0} (-1)^n * 2^n * 3^(-n*(n+1)) * exp( -2/3^(n+1) ) / n!,
%e A326096 where S2 = 0.33802063384093377391547056494398131361711992142768124149541...
%o A326096 (PARI) /* E.g.f.: Sum_{n>=0} ((1+x)^n + 1)^n * x^n/n! */
%o A326096 {a(n) = my(A = sum(m=0,n, ((1+x)^m + 1 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A,n)}
%o A326096 for(n=0,25, print1(a(n),", "))
%o A326096 (PARI) /* E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(x*(1+x)^n) * x^n/n! */
%o A326096 {a(n) = my(A = sum(m=0,n, (1+x +x*O(x^n))^(m^2) * exp(x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
%o A326096 for(n=0,25, print1(a(n),", "))
%Y A326096 Cf. A259208, A326092, A326093, A326094.
%K A326096 nonn
%O A326096 0,2
%A A326096 _Paul D. Hanna_, Jun 05 2019