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A063169 n*A001865(n). 5
1, 6, 51, 568, 7845, 129456, 2485567, 54442368, 1339822377, 36602156800, 1099126705611, 35986038303744, 1275815323139149, 48693140873545728, 1990581237014772375, 86778247940387209216, 4018626330009931930833, 197009947951733259436032, 10193206233792610863520867 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Schenker sums without n-th term.

a(n)/n^n = Q(n) (called Ramanujan's function by Knuth).

Urn, n balls, with replacement: how many selections before a ball is chosen that was chosen already? Expected value is a(n)/n^n.

a(n) is the total number of recurrent elements over all endofunctions on n labeled points.  Sum_{k=1..n} A066324(n,k)*k. - Geoffrey Critzer, Dec 05 2011

REFERENCES

D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, Addison-Wesley, Reading, MA, 1.2.11.3 p. 116

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..150

Marijke van Gans, Schenker sums

FORMULA

a(n) = Sum_{k=0..n-1} n^k * n!/k!.

a(n)/n! = Sum_{k=0..n-1} n^k/k! (first n terms of e^n power series).

E.g.f.: T/(1-T)^2, where T=T(x) is Euler's tree function (see A000169) - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 28 2001

E.g.f.: -LambertW(-x)/(1+LambertW(-x))^2. - Alois P. Heinz, Nov 16 2011

a(n) = A063170(n) - n^n.

a(n) = Sum_{k=1..n} C(n,k) * (n-k)^(n-k) * k^k. - Paul D. Hanna, Jul 04 2013

a(n) ~ n^(n+1/2)*sqrt(Pi/2). - Vaclav Kotesovec, Oct 05 2013

EXAMPLE

a(4) = (1*2*3*4) + 4*(2*3*4) + 4*4*(3*4) + 4*4*4*(4) = 568.

MATHEMATICA

Flatten[Range[0, 20]! CoefficientList[Series[D[1/(1 - y t), y] /. y -> 1, {x, 0, 20}], {x, y}]]

PROG

(UBASIC)

10 for N=1 to 42 : T=N^N : S=0

20 for K=N to 1 step -1 : T/=N : T*=K : S+=T : next K

30 print N, S : next N

(PARI) a(n)=sum(k=1, n, binomial(n, k)*n^(n-k)*k!) /* Michael Somos, Jun 09 2004 */

(PARI) a(n)=sum(k=1, n, binomial(n, k)*(n-k)^(n-k)*k^k) \\ Paul D. Hanna, Jul 04 2013

CROSSREFS

Cf. A001865.

Sequence in context: A057817 A000405 A113352 * A215003 A134525 A125865

Adjacent sequences:  A063166 A063167 A063168 * A063170 A063171 A063172

KEYWORD

nonn,easy,nice

AUTHOR

Marijke van Gans (marijke(AT)maxwellian.demon.co.uk)

STATUS

approved

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Last modified July 31 23:57 EDT 2014. Contains 245103 sequences.