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 A326430 E.g.f.: exp(-1) * Sum_{n>=0} (exp(n*x) + x)^n / n!. 2
 1, 3, 22, 297, 6055, 169431, 6145827, 277912452, 15225719420, 988814989679, 74822364609113, 6505084496930641, 642317112612827029, 71331999557857791694, 8835651007377368848464, 1211946040741011512724559, 182930472229597183037431011, 30216143201862939999461382959, 5435054718681965118312689633935 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS More generally, the following sums are equal: (1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!, (2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!, here, q = exp(x), p = x, r = 1. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA E.g.f.: exp(-1) * Sum_{n>=0} (exp(n*x) + x)^n / n!. E.g.f.: exp(-1) * Sum_{n>=0} exp(n^2*x) * exp( x*exp(n*x) ) / n!. EXAMPLE E.g.f.: A(x) = 1 + 3*x + 22*x^2/2! + 297*x^3/3! + 6055*x^4/4! + 169431*x^5/5! + 6145827*x^6/6! + 277912452*x^7/7! + 15225719420*x^8/8! + 988814989679*x^9/9! + 74822364609113*x^10/10! + ... such that A(x) = exp(-1) * (1 + (exp(x) + x) + (exp(2*x) + x)^2/2! + (exp(3*x) + x)^3/3! + (exp(4*x) + x)^4/4! + (exp(5*x) + x)^5/5! + (exp(6*x) + x)^6/6! + (exp(7*x) + x)^7/7! + (exp(8*x) + x)^8/8! + ...) also, A(x) = exp(-1) * (exp(x) + exp(x)*exp(x*exp(x)) + exp(4*x)*exp(x*exp(2*x))/2! + exp(9*x)*exp(x*exp(3*x))/3! + exp(16*x)*exp(x*exp(4*x))/4! + exp(25*x)*exp(x*exp(5*x))/5! + exp(36*x)*exp(x*exp(6*x))/6! + ...). PROG (PARI) /* Requires appropriate precision */ \p200 {a(n) = my(A = exp(-1) * sum(m=0, n+300, (exp(m*x +x*O(x^n)) + x)^m / m! )); round(n!*polcoeff(A, n))} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A326433. Sequence in context: A244468 A325295 A298693 * A119390 A271848 A144681 Adjacent sequences: A326427 A326428 A326429 * A326431 A326432 A326433 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 09 2019 STATUS approved

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Last modified August 5 02:37 EDT 2024. Contains 374935 sequences. (Running on oeis4.)