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 A326094 E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n!. 5
 1, 5, 27, 185, 1693, 20565, 316375, 5948465, 133579065, 3517749125, 107024710675, 3714813650025, 145570443534805, 6383184292589525, 310815510350462415, 16694390352153656225, 983323269272332915825, 63186890982241624232325, 4409134435821084657726475, 332714992062735780407411225 (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 * r^n/n!, (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*x) * r^n/n!; here, q = (1+x) and p = 4, r = x. 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 = 4, 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 + 4)^n * x^n/n!, E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(4*x*(1+x)^n) * x^n/n!. a(n) = 0 (mod 5) for n > 4. EXAMPLE E.g.f.: A(x) = 1 + 5*x + 27*x^2/2! + 185*x^3/3! + 1693*x^4/4! + 20565*x^5/5! + 316375*x^6/6! + 5948465*x^7/7! + 133579065*x^8/8! + 3517749125*x^9/9! + 107024710675*x^10/10! + ... such that A(x) = 1 + ((1+x) + 4)*x + ((1+x)^2 + 4)^2*x^2/2! + ((1+x)^3 + 4)^3*x^3/3! + ((1+x)^4 + 4)^4*x^4/4! + ((1+x)^5 + 4)^5*x^5/5! + ((1+x)^6 + 4)^6*x^6/6! + ((1+x)^7 + 4)^7*x^7/7! + ... also A(x) = 1 + (1+x)*exp(4*x*(1+x))*x + (1+x)^4*exp(4*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(4*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(4*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(4*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(4*x*(1+x)^6)*x^6/6! + ... PROG (PARI) /* E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n! */ {a(n) = my(A = sum(m=0, n, ((1+x)^m + 4 +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(4*x*(1+x)^n) * x^n/n! */ {a(n) = my(A = sum(m=0, n, (1+x +x*O(x^n))^(m^2) * exp(4*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, A326093. Cf. A326274. Sequence in context: A225309 A231091 A205774 * A232683 A240637 A023811 Adjacent sequences:  A326091 A326092 A326093 * A326095 A326096 A326097 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 21 2019 STATUS approved

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Last modified December 5 12:04 EST 2020. Contains 338947 sequences. (Running on oeis4.)