A090878 -id:A090878 - 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

A128433 Triangle, read by rows, T(n,k) = numerator of the maximum of the k-th Bernstein polynomial of degree n; denominator is A128434.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 27, 3, 27, 1, 1, 256, 216, 216, 256, 1, 1, 3125, 80, 5, 80, 3125, 1, 1, 46656, 37500, 34560, 34560, 37500, 46656, 1, 1, 823543, 5103, 590625, 35, 590625, 5103, 823543, 1, 1, 16777216, 13176688, 1792, 11200000, 11200000, 1792, 13176688, 16777216, 1

Offset: 0

Reinhard Zumkeller, Mar 03 2007

			Triangle begins as:
  1;
  1,        1;
  1,        1,        1;
  1,        4,        4,      1;
  1,       27,        3,     27,        1;
  1,      256,      216,    216,      256,        1;
  1,     3125,       80,      5,       80,     3125,     1;
  1,    46656,    37500,  34560,    34560,    37500, 46656,        1;
  1,   823543,     5103, 590625,       35,   590625,  5103,   823543,        1;
  1, 16777216, 13176688,   1792, 11200000, 11200000,  1792, 13176688, 16777216, 1;

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

Cf. A000169, A000312, A036505, A090878, A128434.

B[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k]*k^k*(n-k)^(n-k)/n^n];
T[n_, k_]= Numerator[B[n, k]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 19 2021 *)

def B(n,k): return 1 if (k==0 or k==n) else binomial(n, k)*k^k*(n-k)^(n-k)/n^n
def T(n,k): return numerator(B(n,k))
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jul 19 2021

T(n,k)/A128434(n,k) = Binomial(n,k) * k^k * (n-k)^(n-k) / n^n.
For n>0: Sum_{k=0..n} T(n,k)/A128434(n,k) = A090878(n)/A036505(n-1).
T(n,n-k) = T(n,k).
T(n,0) = 1.
for n>0: T(n,1)/A128434(n,1) = A000312(n-1)/A000169(n).

A128434 Triangle, read by rows, T(n,k) = denominator of the maximum of the k-th Bernstein polynomial of degree n; numerator is A128433.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 9, 9, 1, 1, 64, 8, 64, 1, 1, 625, 625, 625, 625, 1, 1, 7776, 243, 16, 243, 7776, 1, 1, 117649, 117649, 117649, 117649, 117649, 117649, 1, 1, 2097152, 16384, 2097152, 128, 2097152, 16384, 2097152, 1, 1, 43046721, 43046721, 6561, 43046721, 43046721, 6561, 43046721, 43046721, 1

Offset: 0

Reinhard Zumkeller, Mar 03 2007

			Triangle begins as:
  1;
  1,       1;
  1,       2,      1;
  1,       9,      9,       1;
  1,      64,      8,      64,      1;
  1,     625,    625,     625,    625,       1;
  1,    7776     243,      16,    243,    7776,      1;
  1,  117649, 117649,  117649, 117649,  117649, 117649,       1;
  1, 2097152,  16384, 2097152,    128, 2097152,  16384, 2097152, 1;

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

Cf. A000169, A000312, A036505, A090878, A128433.

B[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k]*k^k*(n-k)^(n-k)/n^n];
T[n_, k_]= Denominator[B[n, k]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 19 2021 *)

def B(n,k): return 1 if (k==0 or k==n) else binomial(n, k)*k^k*(n-k)^(n-k)/n^n
def T(n,k): return denominator(B(n,k))
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jul 19 2021

A128433(n,k)/T(n,k) = binomial(n,k) * k^k * (n-k)^(n-k) / n^n.
For n>0: Sum_{k=0..n} A128433(n,k)/T(n,k) = A090878(n)/A036505(n-1);
T(n, n-k) = T(n,k).
T(n, 0) = T(n, n) = 1.
for n>0: A128433(n,1)/T(n,1) = A000312(n-1)/A000169(n).

A120266 Numerator of Sum_{k=0..n} n^k/k!.

Original entry on oeis.org

2, 5, 13, 103, 1097, 1223, 47273, 556403, 10661993, 7281587, 62929017101, 7218065, 60718862681977, 595953719897, 13324966405463, 247016301114823, 28505097599389815853, 549689343118061, 320305944459287485595917

Offset: 1

Alexander Adamchuk, Jun 30 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 2, 5, 13, 103/3, 1097/12, 1223/5, 47273/72, 556403/315, 10661993/2240, ... = A120266/A214401. - _Petros Hadjicostas_, May 12 2020

Eric Weisstein, Exponential Sum Function.

Denominators are A214401. Cf. also A063170, A090878, A119029, A120267, A214402.

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

a(n) = numerator(Sum_{k=0..n} n^k/k!).

a(n) = A063170(n)/A214402(n) = (n!/A214402(n))*Sum_{k=0..n} n^k/k! for n > 0. - Jonathan Sondow, Jul 16 2012

A036505 Numerator of (n+1)^n/n!.

Original entry on oeis.org

1, 2, 9, 32, 625, 324, 117649, 131072, 4782969, 1562500, 25937424601, 35831808, 23298085122481, 110730297608, 4805419921875, 562949953421312, 48661191875666868481, 91507169819844, 104127350297911241532841, 640000000000000000, 865405750887126927009

Offset: 0

N. J. A. Sloane

Also denominator of Sum_{k=0..n} binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) [Prodinger]. - N. J. A. Sloane, Jul 31 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.

Cf. A036503, A063170.

Cf. A095996 (denominators).

List([0..20], n -> NumeratorRat((n+1)^n/Factorial(n))); # Muniru A Asiru, Feb 12 2018

[Numerator((n+1)^n/Factorial(n)): n in [0..20]]; // Vincenzo Librandi, Sep 10 2013

a:=n -> numer((n+1)^n/factorial(n)):  A036505 := [seq(a(n), n=0..20)]; # Muniru A Asiru, Feb 12 2018

CoefficientList[Series[1/(1 + ProductLog[-x]), {x, 0, 21}], x] // Numerator // Rest (* Jean-François Alcover, Feb 04 2013, after Vladimir Kruchinin *)

my(x='x+O('x^30)); apply(x -> numerator(x), Vec(-1+1/(1+lambertw(-x)))) \\ G. C. Greubel and Michel Marcus, Feb 08 2019

[numerator((n+1)^n/factorial(n)) for n in (0..20)] # G. C. Greubel, Feb 08 2019

a(n) = A090878(n+1)/Sum_{k=0..n+1} (A128433(n+1)/A128434(n+1)). - Reinhard Zumkeller, Mar 03 2007
G.f.: -x*e^(-LambertW(-x))/((LambertW(-x)+1)*LambertW(-x)). - Vladimir Kruchinin, Feb 04 2013
A simpler g.f. is 1/(1 + LambertW(-x)). - Jean-François Alcover, Feb 04 2013

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).

A226931 Numerator of n + Sum(binomial(n,k)(k/n)^k((n-k)/n)^(n-k), k=0..n).

Original entry on oeis.org

3, 9, 53, 231, 5319, 3167, 1296273, 1604979, 64370707, 22906587, 411169704813, 610433321, 424312831956207, 2146177886409, 98731231639051, 12218411169233691, 1112291237880234922707, 2196818399875253, 2619031544578888560315813, 16827894135040576041

Offset: 1

N. J. A. Sloane, Jul 31 2013

			3, 9/2, 53/9, 231/32, 5319/625, 3167/324, 1296273/117649, 1604979/131072, ...

Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7. See xi_2(n).

Denominators are in A036505. Cf. A090878, A063170.

a(n) = numerator(n + sum(k=0, n, binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k))); \\ Michel Marcus, Jun 11 2015

cp's OEIS Frontend

A063170 Schenker sums with n-th term.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Python

UBASIC

Formula

A128433 Triangle, read by rows, T(n,k) = numerator of the maximum of the k-th Bernstein polynomial of degree n; denominator is A128434.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A128434 Triangle, read by rows, T(n,k) = denominator of the maximum of the k-th Bernstein polynomial of degree n; numerator is A128433.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A120266 Numerator of Sum_{k=0..n} n^k/k!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A036505 Numerator of (n+1)^n/n!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

A226931 Numerator of n + Sum(binomial(n,k)(k/n)^k((n-k)/n)^(n-k), k=0..n).