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A326097 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - A(x))^n / n!. 1

%I

%S 1,1,1,11,160,3634,110891,4335204,208768568,12053087736,817245047097,

%T 64036149563110,5723761837812580,577407946342497516,

%U 65153800747494185897,8160944217790837737502,1127265018043808661117840,170726388496282298937412944,28207398922198230159415688865,5061214928838269566809894806406

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

%C More generally, the following sums are equal:

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

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

%C here, q = (1+x) with p = -A(x), r = 1.

%H Paul D. Hanna, <a href="/A326097/b326097.txt">Table of n, a(n) for n = 0..195</a>

%F E.g.f. A(x) satisfies:

%F (1) 1 = Sum_{n>=0} ((1+x)^n - A(x))^n / n!.

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

%F a(4*n+2) = 1 (mod 2), otherwise the terms a(k) are even for k > 3 (conjecture).

%e E.g.f.: A(x) = 1 + x + x^2/2! + 11*x^3/3! + 160*x^4/4! + 3634*x^5/5! + 110891*x^6/6! + 4335204*x^7/7! + 208768568*x^8/8! + 12053087736*x^9/9! + 817245047097*x^10/10! + ...

%e such that

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

%e also

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

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = polcoeff( sum(m=0,#A, ((1+x)^m - Ser(A))^m/m! ),#A-1) ); n!*A[n+1]}

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

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jun 06 2019

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