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Search: id:A063170
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| A063170 |
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Schenker sums with n-th term. |
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5
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| 1, 2, 10, 78, 824, 10970, 176112, 3309110, 71219584, 1727242866, 46602156800, 1384438376222, 44902138752000, 1578690429731402, 59805147699103744, 2428475127395631750, 105224992014096760832, 4845866591896268695010
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Urn, n balls, with replacement: how many selections if we stop after a ball is chosen that was chosen already? Expected value is a(n)/n^n.
a(n) = Integral_{0..infty} exp(-x)*(n+x)^n dx. - Michael Somos May 18 2004
a(n) = Integral_{0..infty} exp(-x)*(1-x/n)^n dx * n^n = A090878(n)/A036505(n-1) * n^n. - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Nov 17 2007
Conjectures: n - A000120(n) is the highest power of 2 dividing a(2*n), 1 is the highest power of 2 dividing a(2*n+1). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Nov 17 2007
EXP-CONV transform of A000312 [From Tilman Neumann (Tilman.Neumann(AT)web.de), Dec 13 2008]
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, Addison-Wesley, p. 123, Exercise Section 1.2.11.3 18.
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LINKS
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Marijke van Gans, Schenker sums
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FORMULA
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a(n) = Sum[k=0..n] n^k n!/k!
a(n)/n! = Sum[k=0..n] n^k/k! (first n+1 terms of e^n power series)
E.g.f.: 1/(1-T)^2, where T=T(x) is Euler's tree function (see A000169).
a(n) = Sum_{k=0..n} binomial(n,k)*(n+k)^k*(-k)^(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 11 2007
Asymptotics of the coefficients : sqrt(Pi*n/2)*n^n. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Jan 25 2008
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EXAMPLE
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e.g. a(4) = (1*2*3*4) + 4*(2*3*4) + 4*4*(3*4) + 4*4*4*(4) + 4*4*4*4
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MAPLE
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seq(simplify(GAMMA(n+1, n)*exp(n)), n=0..20); (Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 21 2005)
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PROGRAM
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(UBASIC) 10 for N=1 to 42: T=N^N: S=T
(UBASIC) 20 for K=N to 1 step -1: T/=N: T*=K: S+=T: next K
(UBASIC) 30 print N, S: next N
(PARI) a(n)=if(n<0, 0, n!*sum(k=0, n, n^k/k!))
(PARI) a(n)=sum(k=0, n, binomial(n, k)*k^k*(n-k)^(n-k)) /* Michael Somos Jun 09 2004 */
(PARI) for(n=0, 17, print1(round(intnum(x=0, [[1], 1], exp(-x)*(n+x)^n)), ", ")) - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Nov 17 2007
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CROSSREFS
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a(n) = A063169(n) + n^n
Cf. A134095.
Cf. A090878, A036505.
Sequence in context: A098692 A138273 A052568 this_sequence A098636 A081363 A100248
Adjacent sequences: A063167 A063168 A063169 this_sequence A063171 A063172 A063173
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KEYWORD
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nonn,easy
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AUTHOR
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Marijke van Gans (marijke(AT)maxwellian.demon.co.uk)
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