A000920 -id:A000920 - cp's OEIS frontend

A131689 Triangle of numbers T(n,k) = k!Stirling2(n,k) = A000142(k)A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320, 0, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880

Offset: 0

Philippe Deléham, Sep 14 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,6,...] where DELTA is the operator defined in A084938; another version of A019538.
See also A019538: version with n > 0 and k > 0. - Philippe Deléham, Nov 03 2008
From Peter Bala, Jul 21 2014: (Start)
T(n,k) gives the number of (k-1)-dimensional faces in the interior of the first barycentric subdivision of the standard (n-1)-dimensional simplex. For example, the barycentric subdivision of the 1-simplex is o--o--o, with 1 interior vertex and 2 interior edges, giving T(2,1) = 1 and T(2,2) = 2.
This triangle is used when calculating the face vectors of the barycentric subdivision of a simplicial complex. Let S be an n-dimensional simplicial complex and write f_k for the number of k-dimensional faces of S, with the usual convention that f_(-1) = 1, so that F := (f_(-1), f_0, f_1,...,f_n) is the f-vector of S. If M(n) denotes the square matrix formed from the first n+1 rows and n+1 columns of the present triangle, then the vector F*M(n) is the f-vector of the first barycentric subdivision of the simplicial complex S (Brenti and Welker, Lemma 2.1). For example, the rows of Pascal's triangle A007318 (but with row and column indexing starting at -1) are the f-vectors for the standard n-simplexes. It follows that A007318*A131689, which equals A028246, is the array of f-vectors of the first barycentric subdivision of standard n-simplexes. (End)
This triangle T(n, k) appears in the o.g.f. G(n, x) = Sum_{m>=0} S(n, m)*x^m with S(n, m) = Sum_{j=0..m} j^n for n >= 1 as G(n, x) = Sum_{k=1..n} (x^k/(1 - x)^(k+2))*T(n, k). See also the Eulerian triangle A008292 with a Mar 31 2017 comment for a rewritten form. For the e.g.f. see A028246 with a Mar 13 2017 comment. - Wolfdieter Lang, Mar 31 2017
T(n,k) = the number of alignments of length k of n strings each of length 1. See Slowinski. An example is given below. Cf. A122193 (alignments of strings of length 2) and A299041 (alignments of strings of length 3). - Peter Bala, Feb 04 2018
The row polynomials R(n,x) are the Fubini polynomials. - Emanuele Munarini, Dec 05 2020
From Gus Wiseman, Feb 18 2022: (Start)
Also the number of patterns of length n with k distinct parts (or with maximum part k), where we define a pattern to be a finite sequence covering an initial interval of positive integers. For example, row n = 3 counts the following patterns:
  (1,1,1)  (1,2,2)  (1,2,3)
           (2,1,2)  (1,3,2)
           (2,2,1)  (2,1,3)
           (1,1,2)  (2,3,1)
           (1,2,1)  (3,1,2)
           (2,1,1)  (3,2,1)
(End)
Regard A048994 as a lower-triangular matrix and divide each term A048994(n,k) by n!, then this is the matrix inverse. Because Sum_{k=0..n} (A048994(n,k) * x^n / n!) = A007318(x,n), Sum_{k=0..n} (A131689(n,k) * A007318(x,k)) = x^n. - Natalia L. Skirrow, Mar 23 2023
T(n,k) is the number of ordered partitions of [n] into k blocks. - Alois P. Heinz, Feb 21 2025

			The triangle T(n,k) begins:
  n\k 0 1    2     3      4       5        6        7        8        9      10 ...
  0:  1
  1:  0 1
  2:  0 1    2
  3:  0 1    6     6
  4:  0 1   14    36     24
  5:  0 1   30   150    240     120
  6:  0 1   62   540   1560    1800      720
  7:  0 1  126  1806   8400   16800    15120     5040
  8:  0 1  254  5796  40824  126000   191520   141120    40320
  9:  0 1  510 18150 186480  834120  1905120  2328480  1451520   362880
  10: 0 1 1022 55980 818520 5103000 16435440 29635200 30240000 16329600 3628800
  ... reformatted and extended. - _Wolfdieter Lang_, Mar 31 2017
From _Peter Bala_, Feb 04 2018: (Start)
T(4,2) = 14 alignments of length 2 of 4 strings of length 1. Examples include
  (i) A -    (ii) A -    (iii) A -
      B -         B -          - B
      C -         - C          - C
      - D         - D          - D
There are C(4,1) = 4 alignments of type (i) with a single gap character - in column 1, C(4,2) = 6 alignments of type (ii) with two gap characters in column 1 and C(4,3) = 4 alignments of type (iii) with three gap characters in column 1, giving a total of 4 + 6 + 4 = 14 alignments. (End)

Vincenzo Librandi, Rows n = 0..100, flattened
Peter Bala, Deformations of the Hadamard product of power series
F. Brenti and V. Welker, f-vectors of barycentric subdivisions, arXiv:math/0606356 [math.CO], Math. Z., 259(4), 849-865, 2008.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Massimo Nocentini, An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation, PhD Thesis, University of Florence, 2019. See Ex. 36.
Mircea Dan Rus, Yet another note on notation, arXiv:2501.08762 [math.HO], 2025. See p. 6.
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
Wikipedia, Barycentric subdivision
Wikipedia, Simplicial complex
Wikipedia, Simplex
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Columns k=0..10 are A000007, A000012, A000918, A001117, A000919, A001118, A000920, A135456, A133068, A133360, A133132,
Case m=1 of the polynomials defined in A278073.
Cf. A000142 (diagonal), A000670 (row sums), A000012 (alternating row sums), A210029 (central terms).
Cf. A001286, A037960, A037961, A037962, A037963.
Cf. A008292, A028246 (o.g.f. and e.g.f. of sums of powers).
Cf. A019538, A122193, A299041.
A version for partitions is A116608, or by maximum A008284.
A version for compositions is A235998, or by maximum A048004.
Classes of patterns:
- A000142 = strict
- A005649 = anti-run, complement A069321
- A019536 = necklace
- A032011 = distinct multiplicities
- A060223 = Lyndon
- A226316 = (1,2,3)-avoiding, weakly A052709, complement A335515
- A296975 = aperiodic
- A345194 = alternating, up/down A350354, complement A350252
- A349058 = weakly alternating
- A351200 = distinct runs
- A351292 = distinct run-lengths
Cf. A007318, A097805, A335456, A335457, A344605, A349050.

function T(n, k)
    if k < 0 || k > n return 0 end
    if n == 0 && k == 0 return 1 end
    k*(T(n-1, k-1) + T(n-1, k))
end
for n in 0:7
    println([T(n, k) for k in 0:n])
end
# Peter Luschny, Mar 26 2020

A131689 := (n,k) -> Stirling2(n,k)*k!: # Peter Luschny, Sep 17 2011
# Alternatively:
A131689_row := proc(n) 1/(1-t*(exp(x)-1)); expand(series(%,x,n+1)); n!*coeff(%,x,n); PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 9 do A131689_row(n) od; # Peter Luschny, Jan 23 2017

t[n_, k_] := k!*StirlingS2[n, k]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014 *)
T[n_, k_] := If[n <= 0 || k <= 0, Boole[n == 0 && k == 0], Sum[(-1)^(i + k) Binomial[k, i] i^(n + k), {i, 0, k}]]; (* Michael Somos, Jul 08 2018 *)

{T(n, k) = if( n<0, 0, sum(i=0, k, (-1)^(k + i) * binomial(k, i) * i^n))};
/* Michael Somos, Jul 08 2018 */

@cached_function
def F(n): # Fubini polynomial
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
for n in (0..9): print(F(n).list()) # Peter Luschny, May 21 2021

T(n,k) = k*(T(n-1,k-1) + T(n-1,k)) with T(0,0)=1. Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A000629(n), A033999(n), A000007(n), A000670(n), A004123(n+1), A032033(n), A094417(n), A094418(n), A094419(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively. [corrected by Philippe Deléham, Feb 11 2013]
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000142(n), A000670(n), A122704(n) for x=-1, 0, 1, 2 respectively. - Philippe Deléham, Oct 09 2007
Sum_{k=0..n} (-1)^k*T(n,k)/(k+1) = Bernoulli numbers A027641(n)/A027642(n). - Peter Luschny, Sep 17 2011
G.f.: F(x,t) = 1 + x*t + (x+x^2)*t^2/2! + (x+6*x^2+6*x^3)*t^3/3! + ... = Sum_{n>=0} R(n,x)*t^n/n!.
The row polynomials R(n,x) satisfy the recursion R(n+1,x) = (x+x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x. - Philippe Deléham, Feb 11 2013
T(n,k) = [t^k] (n! [x^n] (1/(1-t*(exp(x)-1)))). - Peter Luschny, Jan 23 2017
The n-th row polynomial has the form x o x o ... o x (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See also Bala, Example E8. - Peter Bala, Jan 08 2018

A001118 Number of labeled ordered set partitions into 5 parts for n>=1, a(0)=1.

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 5103000, 29607600, 165528000, 901020120, 4809004200, 25292030400, 131542866000, 678330198120, 3474971465400, 17710714165200, 89904730860000, 454951508208120, 2296538629446600, 11570026582092000, 58200094019430000

Offset: 0

N. J. A. Sloane

Previous name: Differences of 0; labeled ordered partitions into 5 parts.
Number of surjections from an n-element set onto a five-element set, with n >= 5. - Mohamed Bouhamida, Dec 15 2007
For n > 0, the number of rows of n colors using exactly five colors. For n=5, the 120 rows are the 120 permutations of ABCDE. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, 2013.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Leipzig, 1911.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (15, -85, 225, -274, 120).

Cf. A001117, A000919, A000920.

Column 5 of A019538, n > 0.

A001118:=-120/(z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1); # Conjectured (correctly) by Simon Plouffe in his 1992 dissertation. Gives sequence except for 5 leading terms.

CoefficientList[Series[(-1-274*x^4+225*x^3-85*x^2+15*x)/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)),{x,0,30}],x] (* Vincenzo Librandi, Apr 11 2012 *)
k=5; Prepend[Table[k!StirlingS2[n,k],{n,1,30}],1] (* Robert A. Russell, Sep 25 2018 *)

a(n) = sum(i=0, 4, (-1)^i*binomial(5, i)*(5-i)^n); \\ Altug Alkan, Dec 04 2015

Vec((-274*x^4 + 225*x^3 - 85*x^2 + 15*x - 1)/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1))+O(x^30)) \\ Stefano Spezia, Oct 16 2018

a(n) = Sum_{i=0..4} (-1)^i*binomial(5, i)*(5-i)^n.
a(n) = [n=0] + 5!*S(n, 5).
E.g.f.: 1 + (e^x-1)^5.
a(n) = 5^n - C(5,4)*4^n + C(5,3)*3^n - C(5,2)*2^n + C(5,1). - Mohamed Bouhamida, Dec 15 2007
G.f.: (-274*x^4 + 225*x^3 - 85*x^2 + 15*x - 1)/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009

Extended with formula and alternate description by Christian G. Bower, Aug 15 1998

Name edited by Harry Richman, Mar 31 2023

A056457 Palindromes using exactly six different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 720, 720, 15120, 15120, 191520, 191520, 1905120, 1905120, 16435440, 16435440, 129230640, 129230640, 953029440, 953029440, 6711344640, 6711344640, 45674188560

Offset: 1

Marks R. Nester

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]

Index entries for linear recurrences with constant coefficients, signature (1,20,-20,-155,155,580,-580,-1044,1044,720,-720).

Cf. A056452, A000920.

k=6; Table[k! StirlingS2[Ceiling[n/2],k],{n,1,30}]

a(n) = 6!*stirling((n+1)\2, 6, 2); \\ Altug Alkan, Sep 25 2018

a(n) = 6! * Stirling2( [(n+1)/2], 6).
G.f.: 720*x^11/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)*(6*x^2-1)). - Colin Barker, Sep 03 2012
G.f.: k!(x^(2k-1)+x^(2k))/Product_{i=1..k}(1-ix^2), where k=6 is the number of symbols. - Robert A. Russell, Sep 25 2018

A135456 Number of surjections from an n-element set onto a seven-element set.

Original entry on oeis.org

5040, 141120, 2328480, 29635200, 322494480, 3162075840, 28805736960, 248619571200, 2060056318320, 16540688324160, 129568848121440, 995210916336000, 7524340159588560, 56163512390086080, 414847224363337920

Offset: 7

Mohamed Bouhamida, Dec 15 2007

G. C. Greubel, Table of n, a(n) for n = 7..1000
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Column k=7 of A019538 and A131689.

Cf. A000918, A000919, A000920, A001117, A001118.

LinearRecurrence[{28, -322, 1960, -6769, 13132, -13068, 5040}, {5040, 141120, 2328480, 29635200, 322494480, 3162075840, 28805736960}, 25] (* G. C. Greubel, Oct 14 2016 *)

a(n) = 7^n -C(7,6)*6^n +C(7,5)*5^n -C(7,4)*4^n +C(7,3)*3^n -C(7,2)*2^n +C(7,1) with n>=7.
a(n) = A000771(n) * 7!. - Max Alekseyev, Nov 12 2009
G.f.: -5040*x^7/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^7. - Ilya Gutkovskiy, Jun 19 2018

More terms from Max Alekseyev, Nov 12 2009

A056313 Number of reversible strings with n beads using exactly six different colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 360, 7560, 95760, 952560, 8217720, 64615680, 476515080, 3355679880, 22837101840, 151449674040, 984573656640, 6302070915840, 39847411326600, 249509384858160, 1550188410555960, 9570844671224760

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent.

			For n=6, the 360 rows are 360 permutations of ABCDEF that do not include any mutual reversals.  Each of the 360 chiral pairs, such as ABCDEF-FEDCBA, is then counted just once. - _Robert A. Russell_, Sep 25 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Index entries for linear recurrences with constant coefficients, signature (19, -117, 81, 1883, -5915, -6615, 53235, -30394, -191744, 264852, 258804, -634248, 43920, 505440, -259200).

Cf. A056308, A056322.
Column 6 of A305621.
Equals (A000920 + A056457) / 2 = A000920 - A305626 = A305626 + A056457.

k=6; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,k,30}] (* Robert A. Russell, Nov 25 2017 *)

a(n) = my(k=6); k!/2*(stirling(n, k, 2) + stirling(ceil(n/2), k, 2)); \\ Altug Alkan, Sep 27 2018

a(n) = A056308(n) - 6*A032122(n) + 15*A032121(n) - 20*A032120(n) + 15*A005418(n+1) - 6.
G.f.: 360*x^6*(8*x^2 - x - 1)*(90*x^7 - 9*x^6 - 29*x^5 - 34*x^4 + 15*x^3 + 9*x^2 - x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)*(6*x^2 - 1)). - Colin Barker, Sep 03 2012
a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=6 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A133068 Number of surjections from an n-element set to an eight-element set.

Original entry on oeis.org

40320, 1451520, 30240000, 479001600, 6411968640, 76592355840, 843184742400, 8734434508800, 86355926616960, 823172919528960, 7621934141203200, 68937160460313600, 611692004959217280, 5342844138794426880, 46061530905262118400, 392795626402384128000

Offset: 8

Mohamed Bouhamida, Dec 16 2007

G. C. Greubel, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Column k=8 of A019538 and A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A135456.

[&+[(-1)^(8-k)*Binomial(8, k)*k^n: k in [1..n]]: n in [8..25]]; // Vincenzo Librandi, Oct 21 2017

CoefficientList[Series[40320*x^8/((x - 1)*(2*x - 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(7*x - 1)*(8*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 20 2017 *)
Table[Sum[(-1)^(8 - k)*Binomial[8, k]*k^n, {k, 1, 8}], {n, 8, 20}] (* G. C. Greubel, Oct 21 2017 *)

x='x+O('x^50); Vec(40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1))) \\ G. C. Greubel, Oct 20 2017

a(n) = Sum_{k=1..8} ((-1)^(8-k)*binomial(8,k)*k^n).
a(n) = A049434(n) * 8!. - Max Alekseyev, Nov 13 2009
G.f.: 40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^8. - Ilya Gutkovskiy, Jun 19 2018

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar

More terms from Max Alekseyev, Nov 13 2009

A133132 Number of surjections from an n-element set to a ten-element set.

Original entry on oeis.org

3628800, 199584000, 6187104000, 142702560000, 2731586457600, 45950224320000, 703098107712000, 10009442963520000, 134672620008326400, 1732015476199008000, 21473732319740064000, 258323865658578720000

Offset: 10

Mohamed Bouhamida, Dec 16 2007

Vincenzo Librandi, Table of n, a(n) for n = 10..1000
Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Column k=10 of A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A049435, A135456.

[10^n-10*9^n+45*8^n-120*7^n+210*6^n-252*5^n+210*4^n-120*3^n+45*2^n-10: n in [10..30]]; // Vincenzo Librandi, Apr 11 2012

With[{nn=30},Drop[CoefficientList[Series[(Exp[x]-1)^10,{x,0,nn}],x] Range[0,nn]!,10]] (* Harvey P. Dale, Sep 01 2016 *)

sum(k=1,10,(-1)^(10-k)*binomial(10,k)*k^n)

a(n) = 10^n-10*9^n+45*8^n-120*7^n+210*6^n-252*5^n+210*4^n-120*3^n+45*2^n-10.
a(n) = A049435(n) * 10!. - Max Alekseyev, Nov 13 2009
G.f.: 3628800*x^10/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x)-1)^10. - Alois P. Heinz, May 17 2016

More terms from Max Alekseyev, Nov 13 2009

Formula corrected by Charles R Greathouse IV, Mar 07 2010

A133360 Number of surjections from an n-element set to a nine-element set.

Original entry on oeis.org

362880, 16329600, 419126400, 8083152000, 130456085760, 1863435974400, 24359586451200, 297846188640000, 3457819037312640, 38528927611574400, 415357755774998400, 4358654246117808000, 44733116259693227520

Offset: 9

Mohamed Bouhamida, Dec 21 2007

Index entries for linear recurrences with constant coefficients, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).

Column k=9 of A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A049447, A135456.

a(n) = Sum_{k=1..9} (-1)^(9-k)*binomial(9,k)*k^n.
a(n) = A049447(n) * 9!. - Max Alekseyev, Nov 12 2009
G.f.: -362880*x^9/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^9. - Ilya Gutkovskiy, Jun 19 2018

More terms from Max Alekseyev, Nov 12 2009

A305626 Number of chiral pairs of rows of n colors with exactly 6 different colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 360, 7560, 95760, 952560, 8217720, 64614960, 476514360, 3355664760, 22837086720, 151449482520, 984573465120, 6302069010720, 39847409421480, 249509368422720, 1550188394120520, 9570844541994120, 58789922099665680, 359629148397511080, 2192484972513916080, 13329510116645202480, 80854267307329446840, 489528474458978944080, 2959252601445086408280, 17866194139995100525080, 107751636988750077294240, 649286502010403671101240

Offset: 1

Robert A. Russell, Jun 06 2018

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

			For a(6) = 360, the chiral pairs are the 6! = 720 permutations of ABCDEF, each paired with its reverse.

Sixth column of A305622.
A056457(n) is number of achiral rows of n colors with exactly 6 different colors.
Cf. A000920, A056313.

k=6; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]

a(n) = 360*(stirling(n, 6, 2) - stirling(ceil(n/2), 6, 2)); \\ Altug Alkan, Sep 26 2018

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=6 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A000920(n) - A056457(n)) / 2.
a(n) = A000920(n) - A056313(n) = A056313(n) - A056457(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=6 colors used.

A371568 Array read by ascending antidiagonals: A(n, k) is the number of paths of length k in Z^n from the origin to points such that x1+x2+...+xn = k with x1,...,xn > 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 6, 1, 0, 0, 6, 14, 1, 0, 0, 0, 36, 30, 1, 0, 0, 0, 24, 150, 62, 1, 0, 0, 0, 0, 240, 540, 126, 1, 0, 0, 0, 0, 120, 1560, 1806, 254, 1, 0, 0, 0, 0, 1800, 8400, 5796, 510, 1

Offset: 1

Shel Kaphan, Mar 28 2024

T(n, k) can also be seen as the number of ordered partitions of k items into n nonempty buckets.
T(n, n) = n!, which is readily seen because to go from the origin to a point in Z^n a distance n away, with at least one step taken in each dimension, the first step can be in any of n dimensions, the second step in any of n-1 dimensions, and so on.
This array is the image of Pascal's triangle A007318 under the Akiyama-Tanigawa transformation. See the Python program. - Peter Luschny, Apr 19 2024

			 n\k 1 2 3  4   5    6     7      8       9       10
  --------------------------------------------------
 1|  1 1 1  1   1    1     1      1       1        1
 2|  0 2 6 14  30   62   126    254     510     1022
 3|  0 0 6 36 150  540  1806   5796   18150    55980
 4|  0 0 0 24 240 1560  8400  40824  186480   818520
 5|  0 0 0  0 120 1800 16800 126000  834120  5103000
 6|  0 0 0  0   0  720 15120 191520 1905120 16435440
 7|  0 0 0  0   0    0  5040 141120 2328480 29635200
 8|  0 0 0  0   0    0     0  40320 1451520 30240000
 9|  0 0 0  0   0    0     0      0  362880 16329600
10|  0 0 0  0   0    0     0      0       0  3628800

Cf. A000918 (n=2), A001117 (n=3), A000919 (n=4), A001118 (n=5), A000920 (n=6).
Cf. A135456 (n=7), A133068 (n=8), A133360 (n=9), A133132 (n=10).
Cf. A371064, A007318.
See A019538 and A131689 for other versions.

A[n_,k_] := Sum[(-1)^(n-i) * i^k * Binomial[n,i], {i,1,n}]

# The Akiyama-Tanigawa algorithm for the binomial generates the rows.
# Adds row(0) = 0^k and column(0) = 0^n.
from math import comb as binomial
def ATBinomial(n, len):
    A = [0] * len
    R = [0] * len
    for k in range(len):
        R[k] = binomial(k, n)
        for j in range(k, 0, -1):
            R[j - 1] = j * (R[j] - R[j - 1])
        A[k] = R[0]
    return A
for n in range(11): print([n], ATBinomial(n, 11))  # Peter Luschny, Apr 19 2024

A(n,k) = Sum_{i=1..n} (-1)^(n-i) * binomial(n,i) * i^k

cp's OEIS Frontend

A131689 Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

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A001118 Number of labeled ordered set partitions into 5 parts for n>=1, a(0)=1.

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A056457 Palindromes using exactly six different symbols.

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A135456 Number of surjections from an n-element set onto a seven-element set.

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A056313 Number of reversible strings with n beads using exactly six different colors.

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A133068 Number of surjections from an n-element set to an eight-element set.

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A133132 Number of surjections from an n-element set to a ten-element set.

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A131689 Triangle of numbers T(n,k) = k!Stirling2(n,k) = A000142(k)A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.