A120266 -id:A120266 - 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

A090878 Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx.

Original entry on oeis.org

2, 5, 26, 103, 2194, 1223, 472730, 556403, 21323986, 7281587, 125858034202, 180451625, 121437725363954, 595953719897, 26649932810926, 3211211914492699, 285050975993898158530, 549689343118061, 640611888918574971191834

Offset: 1

Robert G. Wilson v, Feb 13 2004

Also numerators of e_n(n) where e_n(x) is the exponential sum function exp_n(x) and where denominators are given by either A095996 (largest divisor of n! that is coprime to n) or A036503 (denominator of n^(n-2)/n!). - Gerald McGarvey, Nov 14 2005
a(n) is a multiple of A120266(n) or equals A120266(n), A120266(n) is numerator of Sum_{k=0..n} n^k/k!, the integral = (n-1)!/n^(n-1) * the Sum. - Gerald McGarvey, Apr 17 2008
The integral = (1/n^n)*A063170[n] (Schenker sums with n-th term, Integral_{x>0} exp(-x)*(n+x)^n dx). - Gerald McGarvey, Apr 17 2008
Expected value in the birthday paradox problem.  Let X be a random variable that assigns to each f:{1,2,...,n+1}->{1,2,...,n} the smallest k in {2,3,...,n+1} such that f(k)=f(j) for some j < k.  a(n)/A036505(offset=1) = E(X) the expected value of X.  For n=365 E(X) is (surprising low) approximately 24. - Geoffrey Critzer, May 18 2013
Also numerator of Sum_{k=0..n} binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) [Prodinger]. N. J. A. Sloane, Jul 31 2013

G. C. Greubel, Table of n, a(n) for n = 1..250
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.
Eric Weisstein, Exponential Sum Function

Denominators are in A036505.

Cf. A120266, A063170.

[Numerator((&+[Binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k): k in [0..n]])): n in [1..20]]; // G. C. Greubel, Feb 08 2019

f[n_]:= Integrate[E^(-x)*(1+x/n)^n, {x,0,Infinity}]; Table[Numerator[ f[n]], {n, 1, 20}]
Table[Numerator[1 + Sum[If[k==0,1,Binomial[n,k]*(k/n)^k*((n-k)/n)^(n-k)], {k,0,n-1}]], {n,1,20}] (* G. C. Greubel, Feb 08 2019 *)

vector(20, n, numerator(sum(k=0, n, binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k)))) \\ G. C. Greubel, Feb 08 2019

[numerator(sum(binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) for k in (0..n))) for n in (1..20)] # G. C. Greubel, Feb 08 2019

a(n) = A036505(n-1)*Sum_{k=0..n} (A128433(n)/A128434(n)). - Reinhard Zumkeller, Mar 03 2007

Definition corrected by Gerald McGarvey, Apr 17 2008

A119029 Numerator of Sum_{k=1..n} n^(k-1)/k!.

Original entry on oeis.org

1, 2, 4, 25, 217, 203, 6743, 69511, 1184417, 728102, 5720654791, 601499, 4670663321629, 42568060798, 888330615353, 15438515749903, 1676770323947695709, 30538296012677, 16858207434636875406943

Offset: 1

Alexander Adamchuk, Jul 22 2006

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

			The first few fractions are 1, 2, 4, 25/3, 217/12, 203/5, 6743/72, 69511/315, 1184417/2240, 728102/567, ... = A119029/A214401. - _Petros Hadjicostas_, May 12 2020

Cf. A063170, A090878, A093101, A120266, A120267, A214401 (denominators), A214402.

Numerator[Table[Sum[n^(k-1)/k!,{k,1,n}],{n,1,30}]]

a(n) = numerator(Sum_{k=1..n} n^(k-1)/k!).

a(n) = A120267(n)/n.

A120267 Numerator of Sum_{k=1..n} n^k/k!.

Original entry on oeis.org

1, 4, 12, 100, 1085, 1218, 47201, 556088, 10659753, 7281020, 62927202701, 7217988, 60718623181177, 595952851172, 13324959230295, 247016251998448, 28505095507110827053, 549689328228186, 320305941258100632731917

Offset: 1

Alexander Adamchuk, Jun 30 2006

n divides a(n) and a(n)/n = A119029(n). - Alexander Adamchuk, Oct 08 2006

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

			The first few fractions are 1, 4, 12, 100/3, 1085/12, 1218/5, 47201/72, 556088/315, 10659753/2240, 7281020/567, ... = A120267/A214401. - _Petros Hadjicostas_, May 12 2020

Cf. A063170, A090878, A093101, A119029, A120266, A214401 (denominators), A214402.

Numerator[Table[Sum[n^k/k!, {k,1,n}], {n,1,30}]]

a(n) = numerator(Sum_{k=1..n} n^k/k!).

a(n) = n*A119029(n). - Alexander Adamchuk, Oct 08 2006

Various sections edited by Petros Hadjicostas, May 12 2020~

A214402 Cancellation factor in reducing Sum_{k=0...n} n^k/k! to lowest terms.

Original entry on oeis.org

1, 2, 6, 8, 10, 144, 70, 128, 162, 6400, 22, 6220800, 26, 100352, 182250, 425984, 170, 429981696, 38, 163840000, 13502538, 317194240, 46, 247669456896, 31250, 1417674752, 15943230, 80564191232, 9802, 25076532510720000000, 62, 10737418240, 38196790434, 1241245548544

Offset: 1

Jonathan Sondow, Jul 15 2012

nonn

Michel Marcus, Table of n, a(n) for n = 1..300
Eric Weisstein, Exponential Sum Function.

Cf. A063170, A090878, A093101, A120266, A214401.

Table[n!/Denominator[Sum[n^k/k!, {k, 0, n}]], {n, 1, 30}]

a(n) = n!/denominator(sum(k=0, n, n^k/k!)); \\ Michel Marcus, Apr 20 2021

a(n) = n!/A214401(n).

More terms from Michel Marcus, Apr 20 2021

A214401 Denominator of Sum_{k=0..n} n^k/k!.

Original entry on oeis.org

1, 1, 1, 3, 12, 5, 72, 315, 2240, 567, 1814400, 77, 239500800, 868725, 7175168, 49116375, 2092278988800, 14889875, 3201186852864000, 14849255421, 3783802880000, 3543572316375, 562000363888803840000, 2505147019375, 496358721386591551488

Offset: 1

Jonathan Sondow, Jul 15 2012

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in the current sequence. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

Michel Marcus, Table of n, a(n) for n = 1..150
Eric Weisstein, Exponential Sum Function.

Numerators are A120266.

Cf. also A063170, A090878, A093101, A119029, A120267, A214402.

Denominator[Table[Sum[n^k/k!, {k, 0, n}], {n, 1, 30}]]

a(n) = denominator(sum(k=0, n, n^k/k!)); \\ Michel Marcus, Apr 20 2021

a(n) = n!/A214402(n).

A129977 Numbers m such that A119029(m) = numerator(Sum_{k=1..m} m^(k-1)/k!) is prime.

Original entry on oeis.org

2, 17, 102, 112, 316, 447, 535, 820, 1396, 1475, 1650, 5575, 6486, 6832

Offset: 1

Alexander Adamchuk, Jun 13 2007

For n >= 1, the corresponding primes are A119029(a(n)) = {2, 1676770323947695709, ...}.

a(15) > 10000. - Lucas A. Brown, Apr 01 2021

Lucas A. Brown, A129977.py

Cf. A119029, A120266, A120267.

Do[ f=Numerator[ Sum[ n^(k-1)/k!, {k, 1, n} ] ]; If[ PrimeQ[f], Print[{n,f}] ], {n,1,316} ]
Select[Range[2000],PrimeQ[Numerator[Sum[#^(k-1)/k!,{k,#}]]]&] (* Harvey P. Dale, Jun 15 2019 *)

for( n=1,1000, if( ispseudoprime( numerator( sum( k=1,n,n^(k-1)/k!))), print1(n", "))) \\ M. F. Hasler, Jun 18 2007

Edited and extended (a(6)..a(8)) by M. F. Hasler, Jun 18 2007
More terms from Ryan Propper, Jan 12 2008
Various sections edited by Petros Hadjicostas, May 12 2020
a(12)-a(14) from Lucas A. Brown, Apr 01 2021

A129976 Numbers k such that the numerator of Sum_{j=0..k} k^j/j! is a prime number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 14, 21, 33, 36, 56, 68, 94, 378, 1943, 2389, 2710, 5455, 6804

Offset: 1

Alexander Adamchuk, Jun 13 2007

The corresponding primes are A120266(a(n)) = {2, 5, 13, 103, 1097, 1223, ...}

a(22) > 10^4. - Michael S. Branicky, Apr 27 2025

			Sum_{j=0..4} 4^j/j! = 103/3. The numerator is a prime, hence 4 is in the sequence.

Cf. A120266, A119029, A120267.

Do[ f=Numerator[Sum[n^k/k!,{k,0,n}]]; If[PrimeQ[f], Print[{n,f}]], {n, 1, 378}]

Edited by Stefan Steinerberger, Jul 22 2007
a(17)-a(18) and a(20) from Ryan Propper, Jan 22 2008
a(19) and a(21) from Michael S. Branicky, Apr 26 2025

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A063170 Schenker sums with n-th term.

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A090878 Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx.

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A119029 Numerator of Sum_{k=1..n} n^(k-1)/k!.

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A120267 Numerator of Sum_{k=1..n} n^k/k!.

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A214402 Cancellation factor in reducing Sum_{k=0...n} n^k/k! to lowest terms.

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A214401 Denominator of Sum_{k=0..n} n^k/k!.

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A129977 Numbers m such that A119029(m) = numerator(Sum_{k=1..m} m^(k-1)/k!) is prime.

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