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%I #69 Mar 03 2024 22:07:44
%S 1,6,51,568,7845,129456,2485567,54442368,1339822377,36602156800,
%T 1099126705611,35986038303744,1275815323139149,48693140873545728,
%U 1990581237014772375,86778247940387209216,4018626330009931930833,197009947951733259436032,10193206233792610863520867
%N a(n) = n*A001865(n).
%C Schenker sums without n-th term.
%C a(n)/n^n = Q(n) (called Ramanujan's function by Knuth).
%C Urn, n balls, with replacement: how many selections before a ball is chosen that was chosen already? Expected value is a(n)/n^n.
%C a(n) is the total number of recurrent elements over all endofunctions on n labeled points. a(n) = Sum_{k=1..n} A066324(n,k)*k. - _Geoffrey Critzer_, Dec 05 2011
%D D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, Addison-Wesley, Reading, MA, 1.2.11.3 p. 116
%H Alois P. Heinz, <a href="/A063169/b063169.txt">Table of n, a(n) for n = 1..150</a>
%H 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 Don Zagier, <a href="https://www.youtube.com/watch?v=xVNjOoFNgxY">Partitions, Modular Forms and Moduli Spaces</a>, Youtube video of a lecture at Institut Henri Poincaré, Feb 28 2017.
%F a(n) = Sum_{k=0..n-1} n^k * n!/k!.
%F a(n)/n! = Sum_{k=0..n-1} n^k/k! (first n terms of e^n power series).
%F E.g.f.: T/(1-T)^2, where T=T(x) is Euler's tree function (see A000169) - _Len Smiley_, Nov 28 2001
%F E.g.f.: -LambertW(-x)/(1+LambertW(-x))^2. - _Alois P. Heinz_, Nov 16 2011
%F a(n) = A063170(n) - n^n.
%F a(n) = Sum_{k=1..n} C(n,k) * (n-k)^(n-k) * k^k. - _Paul D. Hanna_, Jul 04 2013
%F a(n) ~ n^(n+1/2)*sqrt(Pi/2). - _Vaclav Kotesovec_, Oct 05 2013
%F a(n) = Sum_{k=1..n} (n!/(n-k)!) * k^2 * n^(n-k-1). - _Brian P Hawkins_, Feb 07 2024
%e a(4) = (1*2*3*4) + 4*(2*3*4) + 4*4*(3*4) + 4*4*4*(4) = 568.
%t Flatten[Range[0, 20]! CoefficientList[Series[D[1/(1 - y t), y] /. y -> 1, {x, 0, 20}], {x, y}]]
%t (* Second program: *)
%t a[n_] := Exp[n]*Gamma[n+1, n] - n^n; Array[a, 19] (* _Jean-François Alcover_, Jan 25 2018 *)
%o (UBASIC)
%o 10 for N=1 to 42 : T=N^N : S=0
%o 20 for K=N to 1 step -1 : T/=N : T*=K : S+=T : next K
%o 30 print N,S : next N
%o (PARI) a(n)=sum(k=1,n,binomial(n,k)*n^(n-k)*k!) /* _Michael Somos_, Jun 09 2004 */
%o (PARI) a(n)=sum(k=1,n,binomial(n,k)*(n-k)^(n-k)*k^k) \\ _Paul D. Hanna_, Jul 04 2013
%o (PARI) a(n)=sum(k=0,n-1,n!/k!*n^k) \\ _Ruud H.G. van Tol_, Jan 14 2023
%o (Python)
%o from math import comb
%o def A063169(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 # _Chai Wah Wu_, Apr 25-26 2023
%Y Cf. A001865, A066324, A219706.
%K nonn,easy,nice
%O 1,2
%A Marijke van Gans (marijke(AT)maxwellian.demon.co.uk)
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