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A063170 Schenker sums with n-th term. 17
1, 2, 10, 78, 824, 10970, 176112, 3309110, 71219584, 1727242866, 46602156800, 1384438376222, 44902138752000, 1578690429731402, 59805147699103744, 2428475127395631750, 105224992014096760832, 4845866591896268695010, 236356356027029797011456 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

Conjectures: The exponent in the power of 2 in the prime factorization of a(n) (its 2-adic valuation) equals 1 if n is odd and equals n - A000120(n) if n is even. - Gerald McGarvey, Nov 17 2007, Jun 29 2012

Amdeberhan, Callan, and Moll (2012) have proved McGarvey's conjectures. - Jonathan Sondow, Jul 16 2012

REFERENCES

D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, Addison-Wesley, p. 123, Exercise Section 1.2.11.3 18.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..385

T. Amdeberhan, D. Callan, and V. Moll, p-adic analysis and combinatorics of truncated exponential sums, preprint, 2012.

T. Amdeberhan, D. Callan and V. Moll, Valuations and combinatorics of truncated exponential sums, INTEGERS 13 (2013), #A21.

Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.

David M. Smith, Geoffrey Smith, Tight Bounds on Information Leakage from Repeated Independent Runs, 2017 IEEE 30th Computer Security Foundations Symposium (CSF).

Marijke van Gans, Schenker sums

Eric Weisstein, Exponential Sum Function.

FORMULA

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).

a(n) = A063169(n) + n^n.

E.g.f.: 1/(1-T)^2, where T=T(x) is Euler's tree function (see A000169).

E.g.f.: 1 / (1 - F), where F = F(x) is the e.g.f. of A003308. - Michael Somos, May 27 2012

a(n) = Sum_{k=0..n} binomial(n,k)*(n+k)^k*(-k)^(n-k). - Vladeta Jovovic, Oct 11 2007

Asymptotics of the coefficients: sqrt(Pi*n/2)*n^n. - N-E. Fahssi, Jan 25 2008

a(n) = A120266(n)*A214402(n) for n > 0. - Jonathan Sondow, Jul 16 2012

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, Nov 17 2007

EXP-CONV transform of A000312. - Tilman Neumann, Dec 13 2008

a(n) = n! * [x^n] exp(n*x)/(1 - x). - Ilya Gutkovskiy, Sep 23 2017

EXAMPLE

E.g. a(4) = (1*2*3*4) + 4*(2*3*4) + 4*4*(3*4) + 4*4*4*(4) + 4*4*4*4

G.f. = 1 + 2*x + 10*x^2 + 78*x^3 + 824*x^4 + 10970*x^5 + 176112*x^6 + ...

MAPLE

seq(simplify(GAMMA(n+1, n)*exp(n)), n=0..20); # Vladeta Jovovic, Jul 21 2005

MATHEMATICA

a[n_] := Round[ Gamma[n+1, n]*Exp[n]]; Table[a[n], {n, 0, 20}] (* Jean-Fran├žois Alcover, Feb 16 2012, after Vladeta Jovovic *)

a[ n_] := If[ n < 1, Boole[n == 0], n! Sum[ n^k / k!, {k, 0, n}]]; (* Michael Somos, Jun 05 2014 *)

a[ n_] := If[ n < 0, 0, n! Normal[ Exp[x] + x O[x]^n] /. x -> n]; (* Michael Somos, Jun 05 2014 *)

PROG

(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, [oo, 1], exp(-x)*(n+x)^n)), ", ")) \\ Gerald McGarvey, Nov 17 2007

CROSSREFS

Cf. A134095, A090878, A036505, A120266, A214402, A219546 (Schenker primes).

Sequence in context: A138273 A301388 A052568 * A098636 A081363 A279908

Adjacent sequences:  A063167 A063168 A063169 * A063171 A063172 A063173

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified July 21 21:14 EDT 2019. Contains 325199 sequences. (Running on oeis4.)