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A063170 Schenker sums with n-th term.

%I A063170 #114 Aug 09 2025 01:35:10
%S A063170 1,2,10,78,824,10970,176112,3309110,71219584,1727242866,46602156800,
%T A063170 1384438376222,44902138752000,1578690429731402,59805147699103744,
%U A063170 2428475127395631750,105224992014096760832,4845866591896268695010,236356356027029797011456
%N A063170 Schenker sums with n-th term.
%C A063170 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.
%C A063170 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
%C A063170 Amdeberhan, Callan, and Moll (2012) have proved McGarvey's conjectures. - _Jonathan Sondow_, Jul 16 2012
%C A063170 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
%D A063170 D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, Addison-Wesley, p. 123, Exercise Section 1.2.11.3 18.
%H A063170 G. C. Greubel, <a href="/A063170/b063170.txt">Table of n, a(n) for n = 0..385</a>
%H A063170 Max Alekseyev, <a href="https://mathoverflow.net/a/486011/231922">Recursion for A063170</a>, answer to question on MathOverflow (2025).
%H A063170 T. Amdeberhan, D. Callan, and V. Moll, <a href="https://web.archive.org/web/20150911180140/http://dauns.math.tulane.edu/~vhm/papers_html/schenker.pdf">p-adic analysis and combinatorics of truncated exponential sums</a>, preprint, 2012.
%H A063170 T. Amdeberhan, D. Callan and V. Moll, <a href="http://www.emis.de/journals/INTEGERS/papers/n21/n21.Abstract.html">Valuations and combinatorics of truncated exponential sums</a>, INTEGERS 13 (2013), #A21.
%H A063170 Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.
%H A063170 Helmut Prodinger, <a href="https://doi.org/10.37236/3309">An identity conjectured by Lacasse via the tree function</a>, Electronic Journal of Combinatorics, 20(3) (2013), #P7.
%H A063170 David M. Smith and Geoffrey Smith, <a href="https://doi.org/10.1109/CSF.2017.18">Tight Bounds on Information Leakage from Repeated Independent Runs</a>, 2017 IEEE 30th Computer Security Foundations Symposium (CSF).
%H A063170 Marijke van Gans, <a href="https://web.archive.org/web/20030420184243/http://www.silicon-alley.com/amp/cbi/seq/schenker.html">Schenker sums</a>
%H A063170 Eric Weisstein, <a href="https://mathworld.wolfram.com/ExponentialSumFunction.html">Exponential Sum Function</a>.
%F A063170 a(n) = Sum_{k=0..n} n^k n!/k!.
%F A063170 a(n)/n! = Sum_{k=0..n} n^k/k!. (First n+1 terms of e^n power series.)
%F A063170 a(n) = A063169(n) + n^n.
%F A063170 E.g.f.: 1/(1-T)^2, where T=T(x) is Euler's tree function (see A000169).
%F A063170 E.g.f.: 1 / (1 - F), where F = F(x) is the e.g.f. of A003308. - _Michael Somos_, May 27 2012
%F A063170 a(n) = Sum_{k=0..n} binomial(n,k)*(n+k)^k*(-k)^(n-k). - _Vladeta Jovovic_, Oct 11 2007
%F A063170 Asymptotics of the coefficients: sqrt(Pi*n/2)*n^n. - _N-E. Fahssi_, Jan 25 2008
%F A063170 a(n) = A120266(n)*A214402(n) for n > 0. - _Jonathan Sondow_, Jul 16 2012
%F A063170 a(n) = Integral_{0..oo} exp(-x) * (n + x)^n dx. - _Michael Somos_, May 18 2004
%F A063170 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
%F A063170 EXP-CONV transform of A000312. - _Tilman Neumann_, Dec 13 2008
%F A063170 a(n) = n! * [x^n] exp(n*x)/(1 - x). - _Ilya Gutkovskiy_, Sep 23 2017
%F A063170 a(n) = (n+1)! - Sum_{k=0..n-1} binomial(n, k)*a(k)*(-k)^(n-k) for n > 0 with a(0) = 1 (see _Max Alekseyev_ link). - _Mikhail Kurkov_, Jan 14 2025
%e A063170 a(4) = (1*2*3*4) + 4*(2*3*4) + 4*4*(3*4) + 4*4*4*(4) + 4*4*4*4.
%e A063170 G.f. = 1 + 2*x + 10*x^2 + 78*x^3 + 824*x^4 + 10970*x^5 + 176112*x^6 + ...
%p A063170 seq(simplify(GAMMA(n+1,n)*exp(n)),n=0..20); # _Vladeta Jovovic_, Jul 21 2005
%t A063170 a[n_] := Round[ Gamma[n+1, n]*Exp[n]]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 16 2012, after _Vladeta Jovovic_ *)
%t A063170 a[ n_] := If[ n < 1, Boole[n == 0], n! Sum[ n^k / k!, {k, 0, n}]]; (* _Michael Somos_, Jun 05 2014 *)
%t A063170 a[ n_] := If[ n < 0, 0, n! Normal[ Exp[x] + x O[x]^n] /. x -> n]; (* _Michael Somos_, Jun 05 2014 *)
%o A063170 (UBASIC)
%o A063170 10 for N=1 to 42: T=N^N: S=T
%o A063170 20 for K=N to 1 step -1: T/=N: T*=K: S+=T: next K
%o A063170 30 print N,S: next N
%o A063170 (PARI) {a(n) = if( n<0, 0, n! * sum( k=0, n, n^k / k!))};
%o A063170 (PARI) {a(n) = sum( k=0, n, binomial(n, k) * k^k * (n - k)^(n - k))}; /* _Michael Somos_, Jun 09 2004 */
%o A063170 (PARI) for(n=0,17,print1(round(intnum(x=0,[oo,1],exp(-x)*(n+x)^n)),", ")) \\ _Gerald McGarvey_, Nov 17 2007
%o A063170 (Python)
%o A063170 from math import comb
%o A063170 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
%Y A063170 Cf. A000312, A134095, A090878, A036505, A120266, A214402, A219546 (Schenker primes).
%K A063170 nonn,easy
%O A063170 0,2
%A A063170 Marijke van Gans (marijke(AT)maxwellian.demon.co.uk)