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A325995
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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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2
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1, 2, 10, 66, 920, 16210, 429672, 13872026, 567140800, 27659216034, 1597736349560, 107173173825322, 8248047045580608, 720418094750908658, 70754645379708307864, 7752521231225769465210, 941123321624950265044352, 125814927060189222097127746, 18423275946242027309159243256, 2940809638786511819597887208906, 509519998644880962798618701505280, 95446391880475967165359534058573202
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OFFSET
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0,2
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COMMENTS
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More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n + p)^n / (1 + p*q^n*r)^(n+k),
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n - p)^n / (1 - p*q^n*r)^(n+k),
for any fixed integer k; here, k = 2 and q = exp(x), p = 1, r = x.
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LINKS
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FORMULA
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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).
E.g.f.: Sum_{n>=0} (n+1) * x^n * (exp(n*x) - 1)^n / (1 - x*exp(n*x))^(n+2).
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).
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.
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).
FORMULAS FOR TERMS.
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.
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.
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EXAMPLE
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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! + ...
such that
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 + ...
also,
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 + ...
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PROG
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(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)}
for(n=0, 25, print1(a(n), ", "))
(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)}
for(n=0, 25, print1(a(n), ", "))
(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 )))}
for(n=0, 25, print1(a(n), ", "))
(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 )))}
for(n=0, 25, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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