login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A063170 Schenker sums with n-th term. 25
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
a(n), for n >= 1, is the number of colored labeled mappings from n points to themselves, where each component is one of two colors. - Steven Finch, Nov 28 2021
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
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.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.
David M. Smith and 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..oo} exp(-x) * (n + x)^n dx. - Michael Somos, May 18 2004
a(n) = Integral_{0..oo} 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
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-François 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
(Python)
from math import comb
def A063170(n): return (sum(comb(n, k)*(n-k)**(n-k)*k**k for k in range(1, (n+1>>1)))<<1) + (0 if n&1 else comb(n, m:=n>>1)*m**n) + (n**n<<1) if n else 1 # Chai Wah Wu, Apr 26 2023
CROSSREFS
Cf. A000312, A134095, A090878, A036505, A120266, A214402, A219546 (Schenker primes).
Sequence in context: A138273 A301388 A052568 * A098636 A081363 A279908
KEYWORD
nonn,easy
AUTHOR
Marijke van Gans (marijke(AT)maxwellian.demon.co.uk)
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 02:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)