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E.g.f. A(x) satisfies: Sum_{n>=0} (exp(n*x) + A(x))^n * x^n / n! = Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!.
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%I #16 Jul 13 2019 17:33:09

%S 1,1,3,19,183,2451,44523,1003955,27132591,862477603,31585560483,

%T 1312509666051,61199375982759,3171627767105747,181223609188848411,

%U 11340823889035215187,772846532507982245727,57069311173494600701763,4546598329397176113578067,389300199395007408056468579,35704214147724534934522349655,3496767016630336049148287129971,364696725110047554147731853993291

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

%C RELATED IDENTITY: 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!.

%H Paul D. Hanna, <a href="/A326550/b326550.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. A(x) allows the following sums to be equal:

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

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

%F (3) B(x) = Sum_{n>=0} exp(n^2*x) * exp(exp(n*x)*x * A(x)) * x^n / n!,

%F (4) B(x) = Sum_{n>=0} exp(n^2*x) * exp(exp(n*x)*x + n*x ) * x^n / n!,

%F where B(x) is the e.g.f. of A326009.

%F a(n) = 1 (mod 2) for n >= 0.

%F a(6*n+1) = 2 (mod 3) for n >= 1;

%F a(6*n+3) = 1 (mod 3) for n >= 0;

%F a(6*n+k) = 0 (mod 3) when k = {0,2,4,5} for n >= 1.

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 183*x^4/4! + 2451*x^5/5! + 44523*x^6/6! + 1003955*x^7/7! + 27132591*x^8/8! + 862477603*x^9/9! + 31585560483*x^10/10! + 1312509666051*x^11/11! + ...

%e such that the following sums are equal

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

%e and

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

%e also

%e (3) B(x) = exp(x*A(x)) + exp(x)*exp(exp(x)*x*A(x))*x + exp(4*x)*exp(exp(2*x)*x*A(x))*x^2/2! + exp(9*x)*exp(exp(3*x)*x*A(x))*x^3/3! + exp(16*x)*exp(exp(4*x)*x*A(x))*x^4/4! + ...

%e and

%e (4) B(x) = exp(x) + exp(x)*exp(exp(x)*x + x)*x + exp(4*x)*exp(exp(2*x)*x + 2*x)*x^2/2! + exp(9*x)*exp(exp(3*x)*x + 3*x)*x^3/3! + exp(16*x)*exp(exp(4*x)*x + 4*x)*x^4/4! + ...

%e where

%e B(x) = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 564*x^4/4! + 7452*x^5/5! + 124126*x^6/6! + 2527646*x^7/7! + 61337576*x^8/8! + 1740438008*x^9/9! + 56893173354*x^10/10! + ... + A326009(n)*x^n/n! + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( sum(m=0,#A, (exp((m+1)*x +x*O(x^#A)) + 1)^m*x^m/m! - (exp(m*x +x*O(x^#A)) + Ser(A))^m*x^m/m! ),#A)); n!*A[n+1]}

%o for(n=0,25,print1(a(n),", "))

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( sum(m=0,#A, exp(m*(m+1)*x + exp(m*x +x*O(x^#A))*x )*x^m/m! - exp(m^2*x + exp(m*x +x*O(x^#A))*x * Ser(A))*x^m/m! ),#A)); n!*A[n+1]}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A326009.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 13 2019