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 A196304 G.f.: x = Sum_{n>=1} a(n)*x^n/(1 + n*(n+1)/2*x)^n. 2
 1, 1, 5, 64, 1587, 65421, 4071178, 357962760, 42379107165, 6512954469625, 1262574678261816, 301690485704179584, 87187147717429037215, 29994563760476311689525, 12119686846920536310216000, 5685713204308826743851247936, 3066004482905684870319668989977 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Compare g.f. to: x = Sum_{n>=1} n^(n-2)*x^n/(1 + n*x)^n, which generates coefficients in the series reversion of x*exp(-x). LINKS Alois P. Heinz, Table of n, a(n) for n = 1..200 FORMULA E.g.f.: x = Sum_{n>=1} a(n)*x^n/(n-1)! * exp(-n*(n+1)/2*x). a(n) = A195737(n)/n for n>=1. a(n) = Sum_{k=1..n-1} (-1)^(k-1)*binomial(n-1,k)*binomial(n+1-k,2)^k*a(n-k) for n>=2. - Jonathan Noel, May 05 2017 EXAMPLE x = x/(1+x) + 1*x^2/(1+3*x)^2 + 5*x^3/(1+6*x)^3 + 64*x^4/(1+10*x)^4 + 1587*x^5/(1+15*x)^5 +...+ a(n)*x^n/(1+n*(n+1)/2*x)^n +... The coefficients a(n) also satisfy: x = x*exp(-x) + 1*x^2/1!*exp(-3*x) + 5*x^3/2!*exp(-6*x) + 64*x^4/3!*exp(-10*x) + 1587*x^5/4!*exp(-15*x) +...+ a(n)*x^n/(n-1)!*exp(-n*(n+1)/2*x) +... MAPLE p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j) -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)): a:= n-> p([i*(i+1)/2\$i=1..n-1]): seq(a(n), n=1..20); # Alois P. Heinz, Dec 03 2015 PROG (PARI) {a(n)=if(n<1, 0, polcoeff(x-sum(m=1, n-1, a(m)*x^m/(1+m*(m+1)/2*x+x*O(x^n))^m), n))} (PARI) {a(n)=if(n<1, 0, (n-1)!*polcoeff(x-sum(m=1, n-1, a(m)*x^m/(m-1)!*exp(-m*(m+1)/2*x+x*O(x^n))), n))} CROSSREFS Cf. A195737, A082157. Sequence in context: A073179 A192558 A179156 * A061684 A061698 A351020 Adjacent sequences: A196301 A196302 A196303 * A196305 A196306 A196307 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 30 2011 STATUS approved

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Last modified September 18 10:16 EDT 2024. Contains 375999 sequences. (Running on oeis4.)