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 A326093 E.g.f.: Sum_{n>=0} ((1+x)^n + 3)^n * x^n/n!. 5
 1, 4, 18, 112, 976, 11424, 169936, 3101032, 67876608, 1746757504, 52034505376, 1771434644544, 68180144988928, 2939951026982272, 140920461751138176, 7457658363325181824, 433145750643704774656, 27464893679743640343552, 1892311278990953945563648, 141074242336048184406390784, 11336870115013701213795557376 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS More generally, the following sums are equal: (1) Sum_{n>=0} (q^n + p)^n * x^n/n!, (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*x) * x^n/n!; here, q = (1+x) and p = 3. In general, let F(x) be a formal power series in x such that F(0)=1, then Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! = Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p); here, F(x) = exp(x), q = 1+x, p = 3, r = x, m = 1. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA E.g.f.: Sum_{n>=0} ((1+x)^n + 3)^n * x^n/n!, E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(3*x*(1+x)^n) * x^n/n!. a(n) = 0 (mod 4) for n > 2. EXAMPLE E.g.f.: A(x) = 1 + 4*x + 18*x^2/2! + 112*x^3/3! + 976*x^4/4! + 11424*x^5/5! + 169936*x^6/6! + 3101032*x^7/7! + 67876608*x^8/8! + 1746757504*x^9/9! + 52034505376*x^10/10! + ... such that A(x) = 1 + ((1+x) + 3)*x + ((1+x)^2 + 3)^2*x^2/2! + ((1+x)^3 + 3)^3*x^3/3! + ((1+x)^4 + 3)^4*x^4/4! + ((1+x)^5 + 3)^5*x^5/5! + ((1+x)^6 + 3)^6*x^6/6! + ((1+x)^7 + 3)^7*x^7/7! + ... also A(x) = 1 + (1+x)*exp(3*x*(1+x))*x + (1+x)^4*exp(3*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(3*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(3*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(3*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(3*x*(1+x)^6)*x^6/6! + ... PROG (PARI) /* E.g.f.: Sum_{n>=0} ((1+x)^n + 3)^n * x^n/n! */ {a(n) = my(A = sum(m=0, n, ((1+x)^m + 3 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) (PARI) /* E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(3*x*(1+x)^n) * x^n/n! */ {a(n) = my(A = sum(m=0, n, (1+x +x*O(x^n))^(m^2) * exp(3*x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A326096, A326092, A326094. Cf. A326273. Sequence in context: A233534 A113356 A062805 * A222584 A294469 A308462 Adjacent sequences:  A326090 A326091 A326092 * A326094 A326095 A326096 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 21 2019 STATUS approved

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Last modified January 22 01:28 EST 2022. Contains 350481 sequences. (Running on oeis4.)