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A324955
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E.g.f.: Sum_{n>=0} x^n * (exp(n*x) + 1)^n / (1 - x*exp(n*x))^(n+1).
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2
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1, 3, 20, 249, 4732, 124185, 4246506, 181846945, 9472372456, 586974323889, 42514726669390, 3548250184843881, 337212466335415980, 36130810233578116537, 4327504309724450845186, 575155450819339262287185, 84280669700080562576116816, 13539565474420162059556816353, 2372630265293612822359985196054, 451491872253364154628541302357529, 92919411414969790075996024292993620
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: Sum_{n>=0} x^n * (exp(n*x) + 1)^n / (1 - x*exp(n*x))^(n+1).
E.g.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (exp(n*x) + exp(k*x))^(n-k).
E.g.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * exp((n-j)*(n-k)*x).
FORMULAS INVOLVING TERMS.
a(n) = Sum_{i=0..n} n!/i! * Sum_{j=0..n-i} binomial(n-i,j) * Sum_{k=0..n-i-j} binomial(n-i-j,k) * (n-i-j)^i * (n-i-k)^i.
a(n) = Sum_{i=0..n} Sum_{j=0..n-i} Sum_{k=0..n-i-j} 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 + 3*x + 20*x^2/2! + 249*x^3/3! + 4732*x^4/4! + 124185*x^5/5! + 4246506*x^6/6! + 181846945*x^7/7! + 9472372456*x^8/8! + 586974323889*x^9/9! + 42514726669390*x^10/10! + ...
such that
A(x) = 1/(1-x) + x*(exp(x) + 1)/(1 - x*exp(x))^2 + x^2*(exp(2*x) + 1)^2/(1 - x*exp(2*x))^3 + x^3*(exp(3*x) + 1)^3/(1 - x*exp(3*x))^4 + x^4*(exp(4*x) + 1)^4/(1 - x*exp(4*x))^5 + x^5*(exp(5*x) + 1)^5/(1 - x*exp(5*x))^6 + x^6*(exp(6*x) + 1)^6/(1 - x*exp(6*x))^7 + x^7*(exp(x)^7 + 1)^7/(1 - x*exp(x)^7)^8 + ...
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PROG
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(PARI) {a(n) = my(A = sum(m=0, n+1, x^m*(exp(m*x +x*O(x^n) ) + 1)^m/(1 - x*exp(m*x +x*O(x^n) ) )^(m+1) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = sum(i=0, n, n!/i! * sum(j=0, n-i, binomial(n-i, j) * sum(k=0, n-i-j, 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, sum(j=0, n-i, sum(k=0, n-i-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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