A003308 -id:A003308 - cp's OEIS frontend

A063170 Schenker sums with n-th term.

1, 2, 10, 78, 824, 10970, 176112, 3309110, 71219584, 1727242866, 46602156800, 1384438376222, 44902138752000, 1578690429731402, 59805147699103744, 2428475127395631750, 105224992014096760832, 4845866591896268695010, 236356356027029797011456

Offset: 0

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

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

			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 + ...

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

G. C. Greubel, Table of n, a(n) for n = 0..385
Max Alekseyev, Recursion for A063170, answer to question on MathOverflow (2025).
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.

Cf. A000312, A134095, A090878, A036505, A120266, A214402, A219546 (Schenker primes).

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

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 *)

{a(n) = if( n<0, 0, n! * sum( k=0, n, n^k / k!))};

{a(n) = sum( k=0, n, binomial(n, k) * k^k * (n - k)^(n - k))}; /* Michael Somos, Jun 09 2004 */

for(n=0,17,print1(round(intnum(x=0,[oo,1],exp(-x)*(n+x)^n)),", ")) \\ Gerald McGarvey, Nov 17 2007

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

10 for N=1 to 42: T=N^N: S=T
20 for K=N to 1 step -1: T/=N: T*=K: S+=T: next K
30 print N,S: next N

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
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

A350149 Triangle read by rows: T(n, k) = n^(n-k)*k!.

Original entry on oeis.org

1, 1, 1, 4, 2, 2, 27, 9, 6, 6, 256, 64, 32, 24, 24, 3125, 625, 250, 150, 120, 120, 46656, 7776, 2592, 1296, 864, 720, 720, 823543, 117649, 33614, 14406, 8232, 5880, 5040, 5040, 16777216, 2097152, 524288, 196608, 98304, 61440, 46080, 40320, 40320

Offset: 0

Robert B Fowler, Dec 27 2021

T(n,k) are the denominators in a double summation power series for the definite integral of x^x. First expand x^x = exp(x*log(x)) = Sum_{n>=0} (x*log(x))^n/n!, then integrate each of the terms to get the double summation for F(x) = Integral_{t=0..x} t^t = Sum_{n>=1} (Sum_{k=0..n-1} (-1)^(n+k+1)*x^n*(log(x))^k/T(n,k)).
This is a definite integral, because lim {x->0} F(x) = 0.
The value of F(1) = 0.78343... = A083648 is known humorously as the Sophomore's Dream (see Borwein et al.).

			Triangle T(n,k) begins:
--------------------------------------------------------------------------
n/k         0        1       2       3      4      5      6      7      8
--------------------------------------------------------------------------
0  |        1,
1  |        1,       1,
2  |        4,       2,      2,
3  |       27,       9,      6,      6,
4  |      256,      64,     32,     24,    24,
5  |     3125,     625,    250,    150,   120,   120,
6  |    46656,    7776,   2592,   1296,   864,   720,   720,
7  |   823543,  117649,  33614,  14406,  8232,  5880,  5040,  5040,
8  | 16777216, 2097152, 524288, 196608, 98304, 61440, 46080, 40320, 40320.
...

Borwein, J., Bailey, D. and Girgensohn, R., Experimentation in Mathematics: Computational Paths to Discovery, A. K. Peters 2004.
William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton NJ 2005.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Sophomore's dream
Wikipedia, Sophomore's dream

Cf. A000312 (first column), A000169 (2nd column), A003308 (3rd column excluding first term), A000142 (main diagonal), A000142 (2nd diagonal excluding first term), A112541 (row sums).
Values of the integral: A083648, A073009.
Cf. A061711, A350297.

A350149:= func< n,k | n^(n-k)*Factorial(k) >;
[A350149(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 31 2022

T := (n, k) -> n^(n - k)*k!:
seq(seq(T(n, k), k = 0..n), n = 0..9); # Peter Luschny, Jan 07 2022

T[n_, k_]:= n^(n-k)*k!; Table[T[n, k], {n, 0,12}, {k,0,n}]//Flatten (* Amiram Eldar, Dec 27 2021 *)

def A350149(n,k): return n^(n-k)*factorial(k)
flatten([[A350149(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 31 2022

T(n, 0) = A000312(n).
T(n, 1) = A000169(n).
T(n, 2) = A003308(n), n >= 2.
Sum_{k=0..n} T(n, k) = A112541(n).
T(n, n) = A000142(n).
T(n, n-1) = A000142(n), n >= 1.
T(n,k) = A061711(n) * (n+1) / A350297(n+1,k). - Robert B Fowler, Jan 11 2022

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A063170 Schenker sums with n-th term.

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A350149 Triangle read by rows: T(n, k) = n^(n-k)*k!.

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