A052762 -id:A052762 - cp's OEIS frontend

A008279 Triangle T(n,k) = n!/(n-k)! (0 <= k <= n) read by rows, giving number of permutations of n things k at a time.

1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 12, 24, 24, 1, 5, 20, 60, 120, 120, 1, 6, 30, 120, 360, 720, 720, 1, 7, 42, 210, 840, 2520, 5040, 5040, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 1, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 362880

Offset: 0

N. J. A. Sloane

Also called permutation coefficients.
Also falling factorials triangle A068424 with column a(n,0)=1 and row a(0,1)=1 otherwise a(0,k)=0, added. - Wolfdieter Lang, Nov 07 2003
The higher-order exponential integrals E(x,m,n) are defined in A163931; for information about the asymptotic expansion of E(x,m=1,n) see A130534. The asymptotic expansions for n = 1, 2, 3, 4, ..., lead to the right hand columns of the triangle given above. - Johannes W. Meijer, Oct 16 2009
The number of injective functions from a set of size k to a set of size n. - Dennis P. Walsh, Feb 10 2011
The number of functions f from {1,2,...,k} to {1,2,...,n} that satisfy f(x) >= x for all x in {1,2,...,k}. - Dennis P. Walsh, Apr 20 2011
T(n,k) = A181511(n,k) for k=1..n-1. - Reinhard Zumkeller, Nov 18 2012
The e.g.f.s enumerating the faces of the permutohedra / permutahedra, Perm(s,t;x) = [e^(sx)-1]/[s-t(e^(sx)-1)], (cf. A090582 and A019538) and the stellahedra / stellohedra, St(s,t;x) = [s e^((s+t)x)]/[s-t(e^(sx)-1)], (cf. A248727) given in Toric Topology satisfy exp[u*d/dt] St(s,t;x) = St(s,u+t;x) = [e^(ux)/(1-u*Perm(s,t;x))]*St(s,t;x), where e^(ux)/(1-uy) is a bivariate e.g.f. for the row polynomials of this entry and A094587. Equivalently, d/dt St = (x+Perm)*St and d/dt Perm = Perm^2, or d/dt log(St) = x + Perm and d/dt log(Perm) = Perm. - Tom Copeland, Nov 14 2016
T(n, k)/n! are the coefficients of the n-th exponential Taylor polynomial, or truncated exponentials, which was proved to be irreducible by Schur. See Coleman link. - Michel Marcus, Feb 24 2020
Given a generic choice of k+2 residues, T(n, k) is the number of meromorphic differentials on the Riemann sphere having a zero of order n and these prescribed residues at its k+2 poles. - Quentin Gendron, Jan 16 2025

			Triangle begins:
  1;
  1,  1;
  1,  2,  2;
  1,  3,  6,   6;
  1,  4, 12,  24,   24;
  1,  5, 20,  60,  120,   120;
  1,  6, 30, 120,  360,   720,    720;
  1,  7, 42, 210,  840,  2520,   5040,   5040;
  1,  8, 56, 336, 1680,  6720,  20160,  40320,   40320;
  1,  9, 72, 504, 3024, 15120,  60480, 181440,  362880,  362880;
  1, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800;
  ...
For example, T(4,2)=12 since there are 12 injective functions f:{1,2}->{1,2,3,4}. There are 4 choices for f(1) and then, since f is injective, 3 remaining choices for f(2), giving us 12 ways to construct an injective function. - _Dennis P. Walsh_, Feb 10 2011
For example, T(5,3)=60 since there are 60 functions f:{1,2,3}->{1,2,3,4,5} with f(x) >= x. There are 5 choices for f(1), 4 choices for f(2), and 3 choices for f(3), giving us 60 ways to construct such a function. - _Dennis P. Walsh_, Apr 30 2011

CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 176; 31st ed., p. 215, Section 3.3.11.1.
Maple V Reference Manual, p. 490, numbperm(n,k).

T. D. Noe, Rows n = 0..100 of triangle, flattened
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013-2014.
V. Buchstaber and T. Panov Toric Topology, arXiv:1210.2368v3 [math.AT], 2014.
R. F. Coleman, On the Galois groups of the exponential Taylor polynomials, L’Enseignement Mathématique 33, 183-189, (1987).
Quentin Gendron and Guillaume Tahar, Isoresidual fibration and resonance arrangements, Lett. Math. Phys. 112, No. 2, Paper No. 33, 36 p. (2022).
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients for k rooks on the 1xn board, all heights 1.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963) 31-41.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Yuriy Shablya and Dmitry Kruchinin, Euler-Catalan's Number Triangle and its Application, Proceedings Book of Micropam (The Mediterranean International Conference of Pure & Applied Mathematics and Related Areas 2018), 212-215.
Dennis Walsh, Notes on no-less functions
Eric Weisstein's World of Mathematics, Falling Factorial
Index entries for "core" sequences

Row sums give A000522.
Cf. A001497, A001498, A136572.
T(n,0)=A000012, T(n,1)=A000027, T(n+1,2)=A002378, T(n,3)=A007531, T(n,4)=A052762, and T(n,n)=A000142.
Cf. A019538, A090582, A094587, A248727.

a008279 n k = a008279_tabl !! n !! k
a008279_row n = a008279_tabl !! n
a008279_tabl = iterate f [1] where
   f xs = zipWith (+) ([0] ++ zipWith (*) xs [1..]) (xs ++ [0])
-- Reinhard Zumkeller, Dec 15 2013, Nov 18 2012

/* As triangle */ [[Factorial(n)/Factorial(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 11 2015

with(combstruct): for n from 0 to 10 do seq(count(Permutation(n),size=m), m = 0 .. n) od; # Zerinvary Lajos, Dec 16 2007
seq(seq(n!/(n-k)!,k=0..n),n=0..10); # Dennis P. Walsh, Apr 20 2011
seq(print(seq(pochhammer(n-k+1,k),k=0..n)),n=0..6); # Peter Luschny, Mar 26 2015

Table[CoefficientList[Series[(1 + x)^m, {x, 0, 20}], x]* Table[n!, {n, 0, m}], {m, 0, 10}] // Grid (* Geoffrey Critzer, Mar 16 2010 *)
Table[ Pochhammer[n - k + 1, k], {n, 0, 9}, {k, 0, n}] // Flatten (* or *)
Table[ FactorialPower[n, k], {n, 0, 9}, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 18 2013, updated Jan 28 2016 *)

{T(n, k) = if( k<0 || k>n, 0, n!/(n-k)!)}; /* Michael Somos, Nov 14 2002 */

{T(n, k) = my(A, p); if( k<0 || k>n, 0, if( n==0, 1, A = matrix(n, n, i, j, x + (i==j)); polcoeff( sum(i=1, n!, if( p = numtoperm(n, i), prod(j=1, n, A[j, p[j]]))), k)))}; /* Michael Somos, Mar 05 2004 */

from math import factorial, isqrt, comb
def A008279(n): return factorial(a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))//factorial(a-n+comb(a+1,2)) # Chai Wah Wu, Nov 13 2024

for n in range(8): [falling_factorial(n,k) for k in (0..n)] # Peter Luschny, Mar 26 2015

E.g.f.: Sum T(n,k) x^n/n! y^k = exp(x)/(1-x*y). - Vladeta Jovovic, Aug 19 2002
Equals A007318 * A136572. - Gary W. Adamson, Jan 07 2008
T(n, k) = n*T(n-1, k-1) = k*T(n-1, k-1)+T(n-1, k) = n*T(n-1, k)/(n-k) = (n-k+1)*T(n, k-1). - Henry Bottomley, Mar 29 2001
T(n, k) = n!/(n-k)! if n >= k >= 0, otherwise 0.
G.f. for k-th column k!*x^k/(1-x)^(k+1), k >= 0.
E.g.f. for n-th row (1+x)^n, n >= 0.
Sum T(n, k)x^k = permanent of n X n matrix a_ij = (x+1 if i=j, x otherwise). - Michael Somos, Mar 05 2004
Ramanujan psi_1(k, x) polynomials evaluated at n+1. - Ralf Stephan, Apr 16 2004
E.g.f.: Sum T(n,k) x^n/n! y^k/k! = e^{x+xy}. - Franklin T. Adams-Watters, Feb 07 2006
The triangle is the binomial transform of an infinite matrix with (1, 1, 2, 6, 24, ...) in the main diagonal and the rest zeros. - Gary W. Adamson, Nov 20 2006
G.f.: 1/(1-x-xy/(1-xy/(1-x-2xy/(1-2xy/(1-x-3xy/(1-3xy/(1-x-4xy/(1-4xy/(1-... (continued fraction). - Paul Barry, Feb 11 2009
T(n,k) = Sum_{j=0..k} binomial(k,j)*T(x,j)*T(y,k-j) for x+y = n. - Dennis P. Walsh, Feb 10 2011
From Dennis P. Walsh, Apr 20 2011: (Start)
E.g.f (with k fixed): x^k*exp(x).
G.f. (with k fixed): k!*x^k/(1-x)^(k+1). (End)
For n >= 2 and m >= 2, Sum_{k=0..m-2} S2(n, k+2)*T(m-2, k) = Sum_{p=0..n-2} m^p. S2(n,k) are the Stirling numbers of the second kind A008277. - Tony Foster III, Jul 23 2019

A130534 Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 11, 6, 1, 24, 50, 35, 10, 1, 120, 274, 225, 85, 15, 1, 720, 1764, 1624, 735, 175, 21, 1, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1

Offset: 0

Philippe Deléham, Aug 09 2007

This triangle is an unsigned version of the triangle of Stirling numbers of the first kind, A008275, which is the main entry for these numbers. - N. J. A. Sloane, Jan 25 2011
Or, triangle T(n,k), 0 <= k <= n, read by rows given by [1,1,2,2,3,3,4,4,5,5,6,6,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.
Reversal of A094638.
Equals A132393*A007318, as infinite lower triangular matrices. - Philippe Deléham, Nov 13 2007
From Johannes W. Meijer, Oct 07 2009: (Start)
The higher order exponential integrals E(x,m,n) are defined in A163931. The asymptotic expansion of the exponential integrals E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + n*(n+1)/x^2 - n*(n+1)*(n+2)/x^3 + ...), see Abramowitz and Stegun. This formula follows from the general formula for the asymptotic expansion, see A163932. We rewrite E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + (n^2+n)/x^2 - (2*n+3*n^2+n^3)/x^3 + (6*n+11*n^2+6*n^3+n^4)/x^3 - ...) and observe that the T(n,m) are the polynomials coefficients in the denominators. Looking at the a(n,m) formula of A028421, A163932 and A163934, and shifting the offset given above to 1, we can write T(n-1,m-1) = a(n,m) = (-1)^(n+m)*Stirling1(n,m), see the Maple program.
The asymptotic expansion leads for values of n from one to eleven to known sequences, see the cross-references. With these sequences one can form the triangles A008279 (right-hand columns) and A094587 (left-hand columns).
See A163936 for information about the o.g.f.s. of the right-hand columns of this triangle.
(End)
The number of elements greater than i to the left of i in a permutation gives the i-th element of the inversion vector. (Skiena-Pemmaraju 2003, p. 69.) T(n,k) is the number of n-permutations that have exactly k 0's in their inversion vector. See evidence in Mathematica code below. - Geoffrey Critzer, May 07 2010
T(n,k) counts the rooted trees with k+1 trunks in forests of "naturally grown" rooted trees with n+2 nodes. This corresponds to sums of coefficients of iterated derivatives representing vectors, Lie derivatives, or infinitesimal generators for flow fields and formal group laws. Cf. links in A139605. - Tom Copeland, Mar 23 2014
A refinement is A036039. - Tom Copeland, Mar 30 2014
From Tom Copeland, Apr 05 2014: (Start)
With initial n=1 and row polynomials of T as p(n,x)=x(x+1)...(x+n-1), the powers of x correspond to the number of trunks of the rooted trees of the "naturally-grown" forest referred to above. With each trunk allowed m colors, p(n,m) gives the number of such non-plane colored trees for the forest with each tree having n+1 vertices.
p(2,m) = m + m^2 = A002378(m) = 2*A000217(m) = 2*(first subdiag of |A238363|).
p(3,m) = 2m + 3m^2 + m^3 = A007531(m+2) = 3*A007290(m+2) = 3*(second subdiag A238363).
p(4,m) = 6m + 11m^2 + 6m^3 + m^4 = A052762(m+3) = 4*A033487(m) = 4*(third subdiag).
From the Joni et al. link, p(n,m) also represents the disposition of n distinguishable flags on m distinguishable flagpoles.
The chromatic polynomial for the complete graph K_n is the falling factorial, which encodes the colorings of the n vertices of K_n and gives a shifted version of p(n,m).
E.g.f. for the row polynomials: (1-y)^(-x).
(End)
A relation to derivatives of the determinant |V(n)| of the n X n Vandermonde matrix V(n) in the indeterminates c(1) thru c(n):
  |V(n)| = Product_{1<=jTom Copeland, Apr 10 2014
From Peter Bala, Jul 21 2014: (Start)
Let M denote the lower unit triangular array A094587 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
  /I_k 0\
  \ 0  M/
having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well defined). See the Example section. (End)
For the relation of this rising factorial to the moments of Viennot's Laguerre stories, see the Hetyei link, p. 4. - Tom Copeland, Oct 01 2015
Can also be seen as the Bell transform of n! without column 0 (and shifted enumeration). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle  T(n,k) begins:
n\k         0        1        2       3       4      5      6     7    8  9 10
n=0:        1
n=1:        1        1
n=2:        2        3        1
n=3:        6       11        6       1
n=4:       24       50       35      10       1
n=5:      120      274      225      85      15      1
n=6:      720     1764     1624     735     175     21      1
n=7:     5040    13068    13132    6769    1960    322     28     1
n=8:    40320   109584   118124   67284   22449   4536    546    36    1
n=9:   362880  1026576  1172700  723680  269325  63273   9450   870   45  1
n=10: 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55  1
[Reformatted and extended by _Wolfdieter Lang_, Feb 05 2013]
T(3,2) = 6 because there are 6 permutations of {1,2,3,4} that have exactly 2 0's in their inversion vector: {1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2}, {2, 1, 3, 4},{2, 3, 1, 4}, {2, 3, 4, 1}. The respective inversion vectors are {0, 0, 1}, {0, 1, 0}, {0, 2, 0}, {1, 0, 0}, {2, 0, 0}, {3, 0, 0}. - _Geoffrey Critzer_, May 07 2010
T(3,1)=11 since there are exactly 11 permutations of {1,2,3,4} with exactly 2 cycles, namely, (1)(234), (1)(243), (2)(134), (2)(143), (3)(124), (3)(142), (4)(123), (4)(143), (12)(34), (13)(24), and (14)(23). - _Dennis P. Walsh_, Jan 25 2011
From _Peter Bala_, Jul 21 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
  / 1          \/1        \/1        \      / 1           \
  | 1  1       ||0 1      ||0 1      |      | 1  1        |
  | 2  2  1    ||0 1 1    ||0 0 1    |... = | 2  3  1     |
  | 6  6  3 1  ||0 2 2 1  ||0 0 1 1  |      | 6 11  6  1  |
  |24 24 12 4 1||0 6 6 3 1||0 0 2 2 1|      |24 50 35 10 1|
  |...         ||...      ||...      |      |...          |
(End)

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 93-94.
Sriram Pemmaraju and Steven Skiena, Computational Discrete Mathematics, Cambridge University Press, 2003, pp. 69-71. [Geoffrey Critzer, May 07 2010]

T. D. Noe, Rows n = 0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251. [From _Johannes W. Meijer_, Oct 07 2009]
A. Chervov, Decomplexification of the Capelli identities and holomorphic factorization, arxiv 1203.5759 [math.QA], Mar 2012. [_Tom Copeland_, Apr 10 2014]
FindStat - Combinatorial Statistic Finder, The number of saliances of the permutation, The number of cycles in the cycle decomposition of a permutation.
Martin Griffiths, Generating Functions for Extended Stirling Numbers of the First Kind, Journal of Integer Sequences, 17 (2014), #14.6.4.
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009.
S. Joni, G. Rota, and B. Sagan, From Sets to Functions: Three Elementary Examples, Discrete Mathematics, vol. 37, no. 2-3, pp. 193-202, 1981. [_Tom Copeland_, Apr 05 2014]
Matthieu Josuat-Verges, A q-analog of Schläfli and Gould identities on Stirling numbers, Preprint, arXiv:1610.02965 [math.CO], 2016.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Lucas Sá and Antonio M. García-García, The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos, arXiv:2104.07647 [hep-th], 2021.
Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.
Dennis Walsh, A short note on unsigned Stirling numbers

See A008275, which is the main entry for these numbers; A094638 (reversed rows).
Diagonals: A000012 A000217 A000914 A001303 A000915 A053567 A112002 A191685. Columns A000142 A000254 A000399 A000454 A000482 A001233 A001234.
From Johannes W. Meijer, Oct 07 2009: (Start)
Row sums equal A000142.
The asymptotic expansions lead to A000142 (n=1), A000142(n=2; minus a(0)), A001710 (n=3), A001715 (n=4), A001720 (n=5), A001725 (n=6), A001730 (n=7), A049388 (n=8), A049389 (n=9), A049398 (n=10), A051431 (n=11), A008279 and A094587.
Cf. A163931 (E(x,m,n)), A028421 (m=2), A163932 (m=3), A163934 (m=4), A163936.
(End)
Cf. A136662.

a130534 n k = a130534_tabl !! n !! k
a130534_row n = a130534_tabl !! n
a130534_tabl = map (map abs) a008275_tabl
-- Reinhard Zumkeller, Mar 18 2013

with(combinat): A130534 := proc(n,m): (-1)^(n+m)*stirling1(n+1,m+1) end proc: seq(seq(A130534(n,m), m=0..n), n=0..10); # Johannes W. Meijer, Oct 07 2009, revised Sep 11 2012
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0 (and shifts the enumeration).
BellMatrix(n -> n!, 9); # Peter Luschny, Jan 27 2016

Table[Table[ Length[Select[Map[ToInversionVector, Permutations[m]], Count[ #, 0] == n &]], {n, 0, m - 1}], {m, 0, 8}] // Grid (* Geoffrey Critzer, May 07 2010 *)
rows = 10;
t = Range[0, rows]!;
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

T(0,0) = 1, T(n,k) = 0 if k > n or if n < 0, T(n,k) = T(n-1,k-1) + n*T(n-1,k). T(n,0) = n! = A000142(n). T(2*n,n) = A129505(n+1). Sum_{k=0..n} T(n,k) = (n+1)! = A000142(n+1). Sum_{k=0..n} T(n,k)^2 = A047796(n+1). T(n,k) = |Stirling1(n+1,k+1)|, see A008275. (x+1)(x+2)...(x+n) = Sum_{k=0..n} T(n,k)*x^k. [Corrected by Arie Bos, Jul 11 2008]
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, respectively. - Philippe Deléham, Nov 13 2007
For k=1..n, let A={a_1,a_2,...,a_k} denote a size-k subset of {1,2,...,n}. Then T(n,n-k) = Sum(Product_{i=1..k} a_i) where the sum is over all subsets A. For example, T(4,1)=50 since 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4 = 50. - Dennis P. Walsh, Jan 25 2011
The preceding formula means T(n,k) = sigma_{n-k}(1,2,3,..,n) with the (n-k)-th elementary symmetric function sigma with the indeterminates chosen as 1,2,...,n. See the Oct 24 2011 comment in A094638 with sigma called there a. - Wolfdieter Lang, Feb 06 2013
From Gary W. Adamson, Jul 08 2011: (Start)
n-th row of the triangle = top row of M^n, where M is the production matrix:
  1, 1;
  1, 2, 1;
  1, 3, 3, 1;
  1, 4, 6, 4, 1;
  ... (End)
Exponential Riordan array [1/(1 - x), log(1/(1 - x))]. Recurrence: T(n+1,k+1) = Sum_{i=0..n-k} (n + 1)!/(n + 1 - i)!*T(n-i,k). - Peter Bala, Jul 21 2014

A050534 Tritriangular numbers: a(n) = binomial(binomial(n,2),2) = n(n+1)(n-1)*(n-2)/8.

Original entry on oeis.org

0, 0, 0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, 2145, 3003, 4095, 5460, 7140, 9180, 11628, 14535, 17955, 21945, 26565, 31878, 37950, 44850, 52650, 61425, 71253, 82215, 94395, 107880, 122760, 139128, 157080, 176715, 198135, 221445, 246753, 274170, 303810, 335790

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

"There are n straight lines in a plane, no two of which are parallel and no three of which are concurrent. Their points of intersection being joined, show that the number of new lines drawn is (1/8)n(n-1)(n-2)(n-3)." (Schmall, 1915).
Several different versions of this sequence are possible, beginning with either one, two or three 0's.
If Y is a 3-subset of an n-set X then, for n>=6, a(n-4) is the number of (n-6)-subsets of X which have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
Number of distinct ways to select 2 pairs of objects from a set of n+1 objects, when order doesn't matter. For example, with n = 3 (4 objects), the 3 possibilities are (12)(34), (13)(24), and (14)(23). - Brian Parsonnet, Jan 03 2012
Partial sums of A027480. - J. M. Bergot, Jul 09 2013
For the set {1,2,...,n}, the sum of the 2 smallest elements of all subsets with 3 elements is a(n) (see Bulut et al. link). - Serhat Bulut, Jan 20 2015
a(n) is also the number of subgroups of S_{n+1} (the symmetric group on n+1 elements) that are isomorphic to D_4 (the dihedral group of order 8). - Geoffrey Critzer, Sep 13 2015
a(n) is the coefficient of x1^(n-3)*x2^2 in exponential Bell polynomial B_{n+1}(x1,x2,...) (number of ways to select 2 pairs among n+1 objects, see above), hence its link with A000292 and A001296 (see formula). - Cyril Damamme, Feb 26 2018
Also the number of 4-cycles in the complete graph K_{n+1}. - Eric W. Weisstein, Mar 13 2018
Number of chiral pairs of colorings of the 4 edges or vertices of a square using n or fewer colors. Each member of a chiral pair is a reflection, but not a rotation, of the other. - Robert A. Russell, Oct 20 2020

			For a(3)=3, the chiral pairs of square colorings are AABC-AACB, ABBC-ACBB, and ABCC-ACCB. - _Robert A. Russell_, Oct 20 2020

Arthur T. Benjamin and Jennifer Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 154.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, page 72.
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5, case k=2.

William A. Tedeschi, Table of n, a(n) for n = 0..10000
Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem, Jan 20 2015.
Alexander Burstein, Sergey Kitaev and Toufik Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A., Vol. 19, No. 2-3 (2008), pp. 27-38.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Frank Harary and Bennet Manvel, On the number of cycles in a graph, Matemat. casop. 21 (1971) 55-63, Theorem 1 for 4-cycles in complete graph.
Louis H. Kauffman, Non-Commutative Worlds-Classical Constraints, Relativity and the Bianchi Identity, arXiv preprint arXiv:1109.1085 [math-ph], 2011. (See Appendix)
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.2.
Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, in Combinatorial Algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
C. N. Schmall, Problem 432, The American Mathematical Monthly, Vol. 22, No. 4 (1915), p. 130.
Eric Weisstein's World of Mathematics, Complete Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Tritriangular Number.
Chai Wah Wu, Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics, arXiv:1407.5663 [quant-ph], 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000332, A033487, A107394, A034827, A210569, Second column of triangle A001498.
Cf. similar sequences listed in A241765.
Cf. (square colorings) A006528 (oriented), A002817 (unoriented), A002411 (achiral),
Row 2 of A325006 (orthoplex facets, orthotope vertices) and A337409 (orthotope edges, orthoplex ridges).
Row 4 of A293496 (cycles of n colors using k or fewer colors).

List([0..40],n->3*Binomial(n+1,4)); # Muniru A Asiru, Mar 20 2018

[3*Binomial(n+1, 4): n in [0..40]]; // Vincenzo Librandi, Feb 14 2015

[seq(binomial(n+1,4)*3,n=0..40)]; # Zerinvary Lajos, Jul 18 2006

Table[Binomial[Binomial[n, 2], 2], {n, 0, 50}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 3, 15}, 40] (* Harvey P. Dale, Dec 14 2011 *)
(* Start from Eric W. Weisstein, Mar 13 2018 *)
Binomial[Binomial[Range[0, 20], 2], 2]
Nest[Binomial[#, 2] &, Range[0, 20], 2]
Nest[PolygonalNumber[# - 1] &, Range[0, 20], 2]
CoefficientList[Series[3 x^3/(1 - x)^5, {x, 0, 20}], x]
(* End *)

a(n)=n*(n+1)*(n-1)*(n-2)/8 \\ Charles R Greathouse IV, Nov 20 2012

x='x+O('x^100); concat([0, 0, 0], Vec(3*x^3/(1-x)^5)) \\ Altug Alkan, Nov 01 2015

[(binomial(binomial(n,2),2)) for n in range(0, 39)] # Zerinvary Lajos, Nov 30 2009

a(n) = 3*binomial(n+1, 4) = 3*A000332(n+1).
From Vladeta Jovovic, May 03 2002: (Start)
Recurrence: a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 3*x^3 / (1-x)^5. (End)
a(n+1) = T(T(n)) - T(n); a(n+2) = T(T(n)+n) where T is A000217. - Jon Perry, Jun 11 2003
a(n+1) = T(n)^2 - T(T(n)) where T is A000217. - Jon Perry, Jul 23 2003
a(n) = T(T(n-1)-1) where T is A000217. - Jon E. Schoenfield, Dec 14 2014
a(n) = 3*C(n, 4) + 3*C(n, 3), for n>3.
From Alexander Adamchuk, Apr 11 2006: (Start)
a(n) = (1/2)*Sum_{k=1..n} k*(k-1)*(k-2).
a(n) = A033487(n-2)/2, n>1.
a(n) = C(n-1,2)*C(n+1,2)/2, n>2. (End)
a(n) = A052762(n+1)/8. - Zerinvary Lajos, Apr 26 2007
a(n) = (4x^4 - 4x^3 - x^2 + x)/2 where x = floor(n/2)*(-1)^n for n >= 0. - William A. Tedeschi, Aug 24 2010
E.g.f.: x^3*exp(x)*(4+x)/8. - Robert Israel, Nov 01 2015
a(n) = Sum_{k=1..n} Sum_{i=1..k} (n-i-1)*(n-k). - Wesley Ivan Hurt, Sep 12 2017
a(n) = A001296(n-1) - A000292(n-1). - Cyril Damamme, Feb 26 2018
Sum_{n>=3} 1/a(n) = 4/9. - Vaclav Kotesovec, May 01 2018
a(n) = A006528(n) - A002817(n) = (A006528(n) - A002411(n)) / 2 = A002817(n) - A002411(n). - Robert A. Russell, Oct 20 2020
Sum_{n>=3} (-1)^(n+1)/a(n) = 32*log(2)/3 - 64/9. - Amiram Eldar, Jan 09 2022
a(n) = Sum_{k=1..2} (-1)^(k+1)*binomial(n,2-k)*binomial(n,2+k). - Gerry Martens, Oct 09 2022

Additional comments from Antreas P. Hatzipolakis, May 03 2002

A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 120, 60, 20, 5, 1, 720, 360, 120, 30, 6, 1, 5040, 2520, 840, 210, 42, 7, 1, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 3628800, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Reinhard Zumkeller, Feb 19 2010

From Wolfdieter Lang, Jun 27 2012: (Start)

T(n-1,k), k=1,...,n-1, gives the number of representative necklaces with n beads (C_N symmetry) of n+1-k distinct colors, say c[1],c[2],...,c[n-k+1], corresponding to the color signature determined by the partition k,1^(n-k) of n. The representative necklaces have k beads of color c[1]. E.g., n=4, k=2: partition 2,1,1, color signature (parts as exponents) c[1]c[1]c[2]c[3], 3=T(3,2) necklaces (write j for color c[j]): cyclic(1123), cyclic(1132) and cyclic(1213). See A212359 for the numbers for general partitions or color signatures. (End)

			Triangle starts:
n\k      1       2      3      4     5    6   7  8  9 10 ...
1        1
2        2       1
3        6       3      1
4       24      12      4      1
5      120      60     20      5     1
6      720     360    120     30     6    1
7     5040    2520    840    210    42    7   1
8    40320   20160   6720   1680   336   56   8  1
9   362880  181440  60480  15120  3024  504  72  9  1
10 3628800 1814400 604800 151200 30240 5040 720 90 10  1
... - _Wolfdieter Lang_, Jun 27 2012

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Index entries for sequences related to factorial numbers.

Row sums give A002627.
Central terms give A006963:
T(2*n-1,n) = A006963(n+1).
T(2*n,n) = A001813(n).
T(2*n,n+1) = A001761(n).
1 < k <= n: T(n,k) = T(n,k-1) / k.
1 <= k <= n: T(n+1,k) = A119741(n,n-k+1).
1 <= k <= n: T(n+1,k+1) = A162995(n,k).
T(n,1) = A000142(n).
T(n,2) = A001710(n) for n>1.
T(n,3) = A001715(n) for n>2.
T(n,4) = A001720(n) for n>3.
T(n,5) = A001725(n) for n>4.
T(n,6) = A001730(n) for n>5.
T(n,7) = A049388(n-7) for n>6.
T(n,8) = A049389(n-8) for n>7.
T(n,9) = A049398(n-9) for n>8.
T(n,10) = A051431(n) for n>9.
T(n,n-7) = A159083(n+1) for n>7.
T(n,n-6) = A053625(n+1) for n>6.
T(n,n-5) = A052787(n) for n>5.
T(n,n-4) = A052762(n) for n>4.
T(n,n-3) = A007531(n) for n>3.
T(n,n-2) = A002378(n-1) for n>2.
T(n,n-1) = A000027(n) for n>1.
T(n,n) = A000012(n).
Cf. A138533, A002627.

a173333 n k = a173333_tabl !! (n-1) !! (k-1)
a173333_row n = a173333_tabl !! (n-1)
a173333_tabl = map fst $ iterate f ([1], 2)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

Table[n!/k!, {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

E.g.f.: (exp(x*y) - 1)/(x*(1 - y)). - Olivier Gérard, Jul 07 2011

T(n,k) = A094587(n,k), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 24, 5, 0, 0, 0, 72, 120, 6, 0, 0, 0, 168, 6720, 360, 7, 0, 0, 0, 360, 935040, 126360, 840, 8, 0, 0, 0, 744, 325061760, 265035240, 1128960, 1680, 9, 0, 0, 0, 1512, 283192323840, 3322711053720, 17160407040, 6510000, 3024, 10

Offset: 1

Alois P. Heinz, May 04 2012

The square diagonal grid graph DG_(n,n) has n^2 = A000290(n) vertices and 2*(n-1)*(2*n-1) = A002943(n-1) edges; see A212208 for example. The chromatic polynomial of DG_(n,n) has n^2+1 = A002522(n) coefficients.

This graph is also called the king graph. - Andrew Howroyd, Jun 25 2017

			Square array A(n,k) begins:
  1,   0,       0,           0,                0, ...
  2,   0,       0,           0,                0, ...
  3,   0,       0,           0,                0, ...
  4,  24,      72,         168,              360, ...
  5, 120,    6720,      935040,        325061760, ...
  6, 360,  126360,   265035240,    3322711053720, ...
  7, 840, 1128960, 17160407040, 2949948395735040, ...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Eric Weisstein's World of Mathematics, King Graph
Wikipedia, Chromatic Polynomial

Columns 1-5 give: A000027, A052762 = 24*A000332, 24*A068250, 24*A068251, 24*A068252.
Rows n=1-16 give: A000007, A000038, 3*A000007, 4*A068293, 5*A068294, 6*A068295, 7*A068296, 8*A068297, 9*A068298, 10*A068299, 11*A068300, 12*A068301, 13*A068302, 14*A068303, 15*A068304, 16*A068305.
Cf. A000290, A002943, A212208, A208021.

A045619 Numbers that are the products of 2 or more consecutive integers.

Original entry on oeis.org

0, 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, 650, 702, 720, 756, 812, 840, 870, 930, 990, 992, 1056, 1122, 1190, 1260, 1320, 1332, 1406, 1482, 1560, 1640, 1680

Offset: 1

Erich Friedman

Erdős and Selfridge proved that, apart from the first term, these are never perfect powers (A001597). - T. D. Noe, Oct 13 2002

Numbers of the form x!/y! with y+1 < x. - Reinhard Zumkeller, Feb 20 2008

			30 is in the sequence as 30 = 5*6 = 5*(5+1). - _David A. Corneth_, Oct 19 2021

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T.D. Noe)
P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois Jour. Math. 19 (1975), 292-301.

Union of A002378, A007531, A052762, A052787, A053625, etc. - R. J. Mathar, Oct 19 2021

Cf. A001597, A000142, A137895, A093449, A064224, A084720, A137899, A137900, A120436, A097889.

maxNum = 1700; lst = {}; For[i = 1, i <= Sqrt[maxNum], i++, j = i + 1; prod = i*j; While[prod < maxNum, AppendTo[lst, prod]; j++; prod *= j]]; lst = Union[lst]

list(lim)=my(v=List([0]),P,k=1,t); while(1, k++; P=binomial('n+k-1,k)*k!; if(subst(P,'n,1)>lim, break); for(n=1,lim, t=eval(P); if(t>lim, next(2)); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Nov 16 2021

import heapq
from sympy import sieve
def aupton(terms, verbose=False):
    p = 6; h = [(p, 2, 3)]; nextcount = 4; aset = {0, 2}
    while len(aset) < terms:
        (v, s, l) = heapq.heappop(h)
        aset.add(v)
        if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} i]")
        if v >= p:
            p *= nextcount
            heapq.heappush(h, (p, 2, nextcount))
            nextcount += 1
        v //= s; s += 1; l += 1; v *= l
        heapq.heappush(h, (v, s, l))
    return sorted(aset)
print(aupton(52)) # Michael S. Branicky, Oct 19 2021

a(n) = A000142(A137911(n))/A000142(A137912(n)-1) for n>1. - Reinhard Zumkeller, Feb 27 2008
Since the oblong numbers (A002378) have relative density of 100%, we have a(n) ~ (n-1) n ~ n^2. - Daniel Forgues, Mar 26 2012
a(n) = n^2 - 2*n^(5/3) + O(n^(4/3)). - Charles R Greathouse IV, Aug 27 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2000
More terms from Reinhard Zumkeller, Feb 27 2008
Incorrect program removed by David A. Corneth, Oct 19 2021

A052787 Product of 5 consecutive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 720, 2520, 6720, 15120, 30240, 55440, 95040, 154440, 240240, 360360, 524160, 742560, 1028160, 1395360, 1860480, 2441880, 3160080, 4037880, 5100480, 6375600, 7893600, 9687600, 11793600, 14250600, 17100720, 20389320, 24165120, 28480320, 33390720

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Appears in Harriot along with the formula (for a different offset) a(n) = n^5 + 10n^4 + 35n^3 + 50n^2 + 24n, see links. - Charles R Greathouse IV, Oct 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Thomas Harriot, Manuscript 6782, p. 77, c. 1599.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 744.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002378, A007531, A052762, A000389, A054559, A173333.

[n*(n-1)*(n-2)*(n-3)*(n-4): n in [0..35]]; // Vincenzo Librandi, May 26 2011

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,Z,Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(numbperm (n,5), n=0..31); # Zerinvary Lajos, Apr 26 2007
G(x):=x^5*exp(x): f[0]:=G(x): for n from 1 to 31 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..31); # Zerinvary Lajos, Apr 05 2009

Times@@@(Partition[Range[-4,35],5,1])  (* Harvey P. Dale, Feb 04 2011 *)

a(n)=120*binomial(n,5) \\ Charles R Greathouse IV, Nov 20 2011

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)=n!/(n-5)!. [Corrected by Philippe Deléham, Dec 12 2003]
a(n) = 120*A000389(n) = 4*A054559(n).
E.g.f.: x^5*exp(x).
Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (-1-n)*a(n)+(-4+n)*a(n+1), a(5)=120}.
O.g.f.: 120*x^5/(-1+x)^6. - R. J. Mathar, Nov 16 2007
For n>5: a(n) = A173333(n,n-5). - Reinhard Zumkeller, Feb 19 2010
a(n) = a(n-1) + 5*A052762(n). - J. M. Bergot, May 30 2012
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=5} 1/a(n) = 1/96.
Sum_{n>=5} (-1)^(n+1)/a(n) = 2*log(2)/3 - 131/288. (End)

More terms from Henry Bottomley, Mar 20 2000

A265609 Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 6, 0, 1, 4, 12, 24, 24, 0, 1, 5, 20, 60, 120, 120, 0, 1, 6, 30, 120, 360, 720, 720, 0, 1, 7, 42, 210, 840, 2520, 5040, 5040, 0, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0

Offset: 0

Peter Luschny, Dec 19 2015

The Pochhammer function is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1.
From Antti Karttunen, Dec 19 2015: (Start)
Apart from the initial row of zeros, if we discard the leftmost column and divide the rest of terms A(n,k) with (n+k) [where k is now the once-decremented column index of the new, shifted position] we get the same array back. See the given recursive formula.
When the numbers in array are viewed in factorial base (A007623), certain repeating patterns can be discerned, at least in a few of the topmost rows. See comment in A001710 and arrays A265890, A265892. (End)
A(n,k) is the k-th moment (about 0) of a gamma (Erlang) distribution with shape parameter n and rate parameter 1. - Geoffrey Critzer, Dec 24 2018

			Square array A(n,k) [where n=row, k=column] is read by ascending antidiagonals as:
A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...
Array starts:
n\k [0  1   2    3     4      5        6         7          8]
--------------------------------------------------------------
[0] [1, 0,  0,   0,    0,     0,       0,        0,         0]
[1] [1, 1,  2,   6,   24,   120,     720,     5040,     40320]
[2] [1, 2,  6,  24,  120,   720,    5040,    40320,    362880]
[3] [1, 3, 12,  60,  360,  2520,   20160,   181440,   1814400]
[4] [1, 4, 20, 120,  840,  6720,   60480,   604800,   6652800]
[5] [1, 5, 30, 210, 1680, 15120,  151200,  1663200,  19958400]
[6] [1, 6, 42, 336, 3024, 30240,  332640,  3991680,  51891840]
[7] [1, 7, 56, 504, 5040, 55440,  665280,  8648640, 121080960]
[8] [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200]
.
Seen as a triangle, T(n, k) = Pochhammer(n - k, k), the first few rows are:
   [0] 1;
   [1] 1, 0;
   [2] 1, 1,  0;
   [3] 1, 2,  2,   0;
   [4] 1, 3,  6,   6,    0;
   [5] 1, 4, 12,  24,   24,    0;
   [6] 1, 5, 20,  60,  120,  120,     0;
   [7] 1, 6, 30, 120,  360,  720,   720,     0;
   [8] 1, 7, 42, 210,  840, 2520,  5040,  5040,     0;
   [9] 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 355.

NIST Digital Library of Mathematical Functions, Pochhammer's Symbol
Index entries for sequences related to factorial numbers

Triangle giving terms only up to column k=n: A124320.
Row 0: A000007, row 1: A000142, row 3: A001710 (from k=1 onward, shifted two terms left).
Column 0: A000012, column 1: A001477, column 2: A002378, columns 3-7: A007531, A052762, A052787, A053625, A159083 (shifted 2 .. 6 terms left respectively, i.e. without the extra initial zeros), column 8: A239035.
Row sums of the triangle: A000522.
A(n, n) = A000407(n-1) for n>0.
2^n*A(1/2,n) = A001147(n).
Cf. also A007623, A008279 (falling factorial), A173333, A257505, A265890, A265892.

for n from 0 to 8 do seq(pochhammer(n,k), k=0..8) od;

Table[Pochhammer[n, k], {n, 0, 8}, {k, 0, 8}]

for n in (0..8): print([rising_factorial(n,k) for k in (0..8)])

(define (A265609 n) (A265609bi (A025581 n) (A002262 n)))
(define (A265609bi row col) (if (zero? col) 1 (* (+ row col -1) (A265609bi row (- col 1)))))
;; Antti Karttunen, Dec 19 2015

A(n,k) = Gamma(n+k)/Gamma(n) for n > 0 and n^k for n=0.
A(n,k) = Sum_{j=0..k} n^j*S1(k,j), S1(n,k) the Stirling cycle numbers A132393(n,k).
A(n,k) = (k-1)!/(Sum_{j=0..k-1} (-1)^j*binomial(k-1, j)/(j+n)) for n >= 1, k >= 1.
A(n,k) = (n+k-1)*A(n,k-1) for k >= 1, A(n,0) = 1. - Antti Karttunen, Dec 19 2015
E.g.f. for row k: 1/(1-x)^k. - Geoffrey Critzer, Dec 24 2018
A(n, k) = FallingFactorial(n + k - 1, k). - Peter Luschny, Mar 22 2022
G.f. for row n as a continued fraction of Stieltjes type: 1/(1 - n*x/(1 - x/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 3*x/(1 - ... ))))))). See Wall, Chapter XVIII, equation 92.5. Cf. A226513. - Peter Bala, Aug 27 2023

cp's OEIS Frontend

A217056 Highly composite numbers (A002182) which are the product of 4 consecutive integers (A052762).

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A008279 Triangle T(n,k) = n!/(n-k)! (0 <= k <= n) read by rows, giving number of permutations of n things k at a time.

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A130534 Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.

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A050534 Tritriangular numbers: a(n) = binomial(binomial(n,2),2) = n*(n+1)*(n-1)*(n-2)/8.

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A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

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A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).

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A050534 Tritriangular numbers: a(n) = binomial(binomial(n,2),2) = n(n+1)(n-1)*(n-2)/8.

A011915 a(n) = floor(n(n-1)(n-2)*(n-3)/5).