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E.g.f.: Sum_{n>=0} (n+1) * x^n * (exp(n*x) + 1)^n / (1 + x*exp(n*x))^(n+2).
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%I #18 Jul 06 2019 09:40:06

%S 1,2,10,66,920,16210,429672,13872026,567140800,27659216034,

%T 1597736349560,107173173825322,8248047045580608,720418094750908658,

%U 70754645379708307864,7752521231225769465210,941123321624950265044352,125814927060189222097127746,18423275946242027309159243256,2940809638786511819597887208906,509519998644880962798618701505280,95446391880475967165359534058573202

%N E.g.f.: Sum_{n>=0} (n+1) * x^n * (exp(n*x) + 1)^n / (1 + x*exp(n*x))^(n+2).

%C More generally, the following sums are equal:

%C (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n + p)^n / (1 + p*q^n*r)^(n+k),

%C (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n - p)^n / (1 - p*q^n*r)^(n+k),

%C for any fixed integer k; here, k = 2 and q = exp(x), p = 1, r = x.

%F E.g.f.: Sum_{n>=0} (n+1) * x^n * (exp(n*x) + 1)^n / (1 + x*exp(n*x))^(n+2).

%F E.g.f.: Sum_{n>=0} (n+1) * x^n * (exp(n*x) - 1)^n / (1 - x*exp(n*x))^(n+2).

%F E.g.f.: Sum_{n>=0} (n+1) * x^n * Sum_{k=0..n} binomial(n,k) * ( exp(n*x) - exp(k*x) )^(n-k).

%F E.g.f.: Sum_{n>=0} (n+1) * x^n * Sum_{k=0..n} binomial(n,k) * ( exp(n*x) + exp(k*x) )^(n-k) * (-1)^k.

%F E.g.f.: Sum_{n>=0} (n+1) * x^n * Sum_{k=0..n} binomial(n,k) * (-1)^k * Sum_{j=0..n-k} binomial(n-k,j) * exp((n-k)*(n-j)*x).

%F FORMULAS FOR TERMS.

%F a(n) = Sum_{i=0..n} (n-i+1) * n!/i! * Sum_{j=0..n-i} binomial(n-i,j) * Sum_{k=0..n-i-j} (-1)^k * binomial(n-i-j,k) * (n-i-j)^i * (n-i-k)^i.

%F a(n) = Sum_{i=0..n} (n-i+1) * Sum_{j=0..n-i} Sum_{k=0..n-i-j} (-1)^k * n!*(n-i)! / ((n-i-j-k)! * i!*j!*k!) * (n-i-j)^i * (n-i-k)^i.

%e E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 66*x^3/3! + 920*x^4/4! + 16210*x^5/5! + 429672*x^6/6! + 13872026*x^7/7! + 567140800*x^8/8! + 27659216034*x^9/9! + 1597736349560*x^10/10! + ...

%e such that

%e A(x) = 1/(1+x)^2 + 2*x*(exp(x) + 1)/(1 + x*exp(x))^3 + 3*x^2*(exp(2*x) + 1)^2/(1 + x*exp(2*x))^4 + 4*x^3*(exp(3*x) + 1)^3/(1 + x*exp(3*x))^5 + 5*x^4*(exp(4*x) + 1)^4/(1 + x*exp(4*x))^6 + 6*x^5*(exp(5*x) + 1)^5/(1 + x*exp(5*x))^7 + 7*x^6*(exp(6*x) + 1)^6/(1 + x*exp(6*x))^8 + 8*x^7*(exp(x)^7 + 1)^7/(1 + x*exp(x)^7)^9 + ...

%e also,

%e A(x) = 1/(1-x)^2 + 2*x*(exp(x) - 1)/(1 - x*exp(x))^3 + 3*x^2*(exp(2*x) - 1)^2/(1 - x*exp(2*x))^4 + 4*x^3*(exp(3*x) - 1)^3/(1 - x*exp(3*x))^5 + 5*x^4*(exp(4*x) - 1)^4/(1 - x*exp(4*x))^6 + 6*x^5*(exp(5*x) - 1)^5/(1 - x*exp(5*x))^7 + 7*x^6*(exp(6*x) - 1)^6/(1 - x*exp(6*x))^8 + 8*x^7*(exp(x)^7 - 1)^7/(1 - x*exp(x)^7)^9 + ...

%o (PARI) {a(n) = my(A = sum(m=0, n+1, (m+1) * x^m*(exp(m*x +x*O(x^n) ) + 1)^m/(1 + x*exp(m*x +x*O(x^n) ) )^(m+2) )); n!*polcoeff(A, n)}

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

%o (PARI) {a(n) = my(A = sum(m=0, n+1, (m+1) * x^m*(exp(m*x +x*O(x^n) ) - 1)^m/(1 - x*exp(m*x +x*O(x^n) ) )^(m+2) )); n!*polcoeff(A, n)}

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

%o (PARI) {a(n) = sum(i=0, n, (n-i+1) * n!/i! * sum(j=0, n-i, binomial(n-i, j) * sum(k=0, n-i-j, (-1)^k * binomial(n-i-j, k) * (n-i-j)^i * (n-i-k)^i )))}

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

%o (PARI) {a(n) = sum(i=0, n, (n-i+1) * sum(j=0, n-i, sum(k=0, n-i-j, (-1)^j*n!*(n-i)!/((n-i-j-k)!*i!*j!*k!) * (n-i-j)^i * (n-i-k)^i )))}

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

%Y Cf. A324954, A325996.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 01 2019