|
| |
|
|
A063170
|
|
Schenker sums with n-th term.
|
|
8
|
|
|
|
1, 2, 10, 78, 824, 10970, 176112, 3309110, 71219584, 1727242866, 46602156800, 1384438376222, 44902138752000, 1578690429731402, 59805147699103744, 2428475127395631750, 105224992014096760832, 4845866591896268695010
(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
|
Table of n, a(n) for n=0..17.
T. Amdeberhan, D. Callan, and V. Moll, p-adic analysis and combinatorics of truncated exponential sums, preprint, 2012.
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]
|
|
|
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
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-François Alcover, Feb 16 2012, after Vladeta Jovovic *)
|
|
|
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, [[1], 1], exp(-x)*(n+x)^n)), ", ")) - Gerald McGarvey, Nov 17 2007
|
|
|
CROSSREFS
|
Cf. A134095.
Cf. A090878, A036505, A120266, A214402.
Cf. A219546 (Schenker primes).
Sequence in context: A098692 A138273 A052568 * A098636 A081363 A100248
Adjacent sequences: A063167 A063168 A063169 * A063171 A063172 A063173
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Marijke van Gans (marijke(AT)maxwellian.demon.co.uk)
|
|
|
STATUS
|
approved
|
| |
|
|
