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A326434 E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.

%I A326434 #12 Jul 20 2019 16:33:43
%S A326434 1,4,47,895,24450,887803,40818505,2297393888,154381810471,
%T A326434 12149510583583,1102672816721422,113974516318639363,
%U A326434 13277046519634998953,1727765194711759098324,249264545884060054668295,39606622952407779396832791,6891271396238954765341535650,1306288225868329080524305347859,268542657134280438710389415260401,59628381166607045580114829853101712
%N A326434 E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.
%C A326434 More generally, the following sums are equal:
%C A326434 (1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
%C A326434 (2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
%C A326434 here, q = exp(x), p = 2, r = 1.
%F A326434 E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.
%F A326434 E.g.f.: exp(-3) * Sum_{n>=0} exp(n^2*x) * exp( 2*exp(n*x) ) / n!.
%F A326434 FORMULAS FOR TERMS.
%F A326434 a(3*n+1) = 0 (mod 2), a(3*n) = 1 (mod 2), and a(3*n+2) = 1 (mod 2) for n >= 0.
%e A326434 E.g.f.: A(x) = 1 + 4*x + 47*x^2/2! + 895*x^3/3! + 24450*x^4/4! + 887803*x^5/5! + 40818505*x^6/6! + 2297393888*x^7/7! + 154381810471*x^8/8! + 12149510583583*x^9/9! + 1102672816721422*x^10/10! + ...
%e A326434 such that
%e A326434 A(x) = exp(-3) * (1 + (exp(x) + 2) + (exp(2*x) + 2)^2/2! + (exp(3*x) + 2)^3/3! + (exp(4*x) + 2)^4/4! + (exp(5*x) + 2)^5/5! + (exp(6*x) + 2)^6/6! + ...)
%e A326434 also
%e A326434 A(x) = exp(-3) * (exp(2) + exp(x)*exp(2*exp(x)) + exp(4*x)*exp(2*exp(2*x))/2! + exp(9*x)*exp(2*exp(3*x))/3! + exp(16*x)*exp(2*exp(4*x))/4! + exp(25*x)*exp(2*exp(5*x))/5! + exp(36*x)*exp(2*exp(6*x))/6! + ...).
%o A326434 (PARI) /* Requires suitable precision */
%o A326434 \p200
%o A326434 Vec(round(serlaplace( exp(-3) * sum(n=0, 500, (exp(n*x +O(x^31)) + 2)^n/n! ) )))
%Y A326434 Cf. A326600, A020557, A326433, A326435, A326436, A326437.
%K A326434 nonn
%O A326434 0,2
%A A326434 _Paul D. Hanna_, Jul 11 2019