A008279 -id:A008279 - cp's OEIS frontend

A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

1, 2, 4, 1, 4, 32, 38, 12, 1, 8, 208, 652, 576, 188, 24, 1, 16, 1280, 9080, 16944, 12052, 3840, 580, 40, 1, 32, 7744, 116656, 412800, 540080, 322848, 98292, 16000, 1390, 60, 1, 64, 46592, 1446368, 9196992, 20447056, 20453376, 10564304, 3047520, 511392, 50400

Offset: 1

N. J. A. Sloane, Dec 21 2002

A generalization of the Stirling2 numbers S_{1,1} from A008277.
The g.f. for column k=2*K is (x^K)*pe(K,x)*d(k,x) and for k=2*K+1 it is (x^K)*po(K,x)*2*(K+1)*K*d(k,x), K>= 1, with d(k,x) := 1/product(1-p*(p-1)*x,p=2..k) and the row polynomials pe(n,x) := sum(A089275(n,m)*x^m,m=0..n-1) and po(n,x) := sum(A089276(n,m)*x^m,m=0..n-1). - Wolfdieter Lang, Nov 07 2003
The formula for the k-th column sequence is given in A089511.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_2 (the disjoint union of n copies of the complete graph K_2). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k | 2    3    4    5    6   7   8
= = = = = = = = = = = = = = = = = =
  1 | 1
  2 | 2    4    1
  3 | 4   32   38   12    1
  4 | 8  208  652  576  188  24   1
...
Graph coloring interpretation of T(2,3) = 4: The graph 2K_2 is 2 copies of K_2, the complete graph on 2 vertices:
o---o  o---o
a   b  c   d
The four 3-colorings of 2K_2 are ac|b|d, ad|b|c, bc|a|d and bd|a|c. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
Steve Butler, Fan Chung, Jay Cummings, R. L. Graham, Juggling card sequences, arXiv:1504.01426 [math.CO], 2015.
Leonard Carlitz, On Arrays of Numbers, Am. J. Math., 54,4 (1932) 739-752. [Eqs. (3) and (4) with lambda = 0, mu = 2, a_{n,k-1} = a(n, k).- _Wolfdieter Lang_, Jan 30 2020 ]
P. Codara, O. M. D’Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers arXiv:1308.1700v1 [cs.DM]
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
S.-M. Ma, T. Mansour, M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

Row sums give A020556. Triangle S_{1, 1} = A008277, S_{2, 1} = A008297 (ignoring signs), S_{3, 1} = A035342, S_{3, 2} = A078740, S_{3, 3} = A078741. A090214 (S_{4,4}).
The column sequences are A000079(n-1)(powers of 2), 4*A016129(n-2), A089271, 12*A089272, A089273, etc.
Main diagonal is A217900.
Cf. A071951 (Legendre-Stirling triangle).
Cf. A122193, A055203.

# Note that the function implements the full triangle because it can be
# much better reused and referenced in this form.
A078739 := proc(n,k) local r;
add((-1)^(n-r)*binomial(n,r)*combinat[stirling2](n+r,k),r=0..n) end:
# Displays the truncated triangle from the definition:
seq(print(seq(A078739(n,k),k=2..2*n)),n=1..6); # Peter Luschny, Mar 25 2011

t[n_, k_] := Sum[(-1)^(n-r)*Binomial[n, r]*StirlingS2[n+r, k], {r, 0, n}]; Table[t[n, k], {n, 1, 7}, {k, 2, 2*n}] // Flatten (* Jean-François Alcover, Apr 11 2013, after Peter Luschny *)

a(n, k) = sum(binomial(k-2+p, p)*A008279(2, p)*a(n-1, k-2+p), p=0..2) if 2 <= k <= 2*n for n>=1, a(1, 2)=1; else 0. Here A008279(2, p) gives the third row (k=2) of the augmented falling factorial triangle: [1, 2, 2] for p=0, 1, 2. From eq.(21) with r=2 of the Blasiak et al. paper.
a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*A008279(p, 2)^n, p=2..k) for 2 <= k <= 2*n, n>=1. From eq.(19) with r=2 of the Blasiak et al. paper.
a(n, k) = sum(A071951(n, j)*A089503(j, 2*j-k+1), j=ceiling(k/2)..min(n, k-1)), 1<=n, 2<=k<=2n; relation to Legendre-Stirling triangle. Wolfdieter Lang, Dec 01 2003
a(n, k) = A122193(n,k)*2^n/k! - Peter Luschny, Mar 25 2011
E^n = sum_{k=2}^(2n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^2d^2/dx^2.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*sum {k = 0..inf} (k*(k-1))^n*x^k/k!. - Peter Bala, Aug 15 2013

More terms from Wolfdieter Lang, Nov 07 2003

A144084 T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 72, 96, 24, 1, 25, 200, 600, 600, 120, 1, 36, 450, 2400, 5400, 4320, 720, 1, 49, 882, 7350, 29400, 52920, 35280, 5040, 1, 64, 1568, 18816, 117600, 376320, 564480, 322560, 40320

Offset: 0

Abdullahi Umar, Sep 10 2008, Sep 30 2008

T(n,k) is also the number of elements in the Green's J equivalence classes in the symmetric inverse monoid, I sub n.
T(n,k) is also the number of ways to place k nonattacking rooks on an n X n chessboard. It can be obtained by performing P(n,k) permutations of n-columns over each C(n,k) combination of n-rows for the given k-rooks. The rule is also applicable for unequal (m X n) sized rectangular boards. - Antal Pinter, Nov 12 2014
Rows also give the coefficients of the matching-generating polynomial of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017
Rows also give the coefficients of the independence polynomial of the n X n rook graph and clique polynomial of the n X n rook complement graph. - Eric W. Weisstein, Jun 13 and Sep 14 2017
T(n,k) is the number of increasing subsequences of length n-k over all permutations of [n]. - Geoffrey Critzer, Jan 08 2023

			T(3,1) = 9 because there are exactly 9 partial bijections (on a 3-element set) of height 1, namely: (1)->(1), (1)->(2), (1)->(3), (2)->(1), (2)->(2), (2)->(3), (3)->(1), (3)->(2), (3)->(3).
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   2;
  1,  9,  18,    6;
  1, 16,  72,   96,   24;
  1, 25, 200,  600,  600,  120;
  1, 36, 450, 2400, 5400, 4320, 720;
  ...

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, page 61.
J. M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995).
Vaclav Kotesovec, Non-attacking chess pieces, 6th ed. (2013), p. 216, p. 218.

Alois P. Heinz, Rows n = 0..140, flattened
Wayne A. Johnson, Exponential Hilbert series of equivariant embeddings, arXiv:1804.04943 [math.RT], 2018.
W. D. Munn, The characters of the symmetric inverse semigroup, Proc. Cambridge Philos. Soc. 53 (1957), 13-18.
Eric Weisstein's World of Mathematics, Clique Polynomial
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Rook Complement Graph
Eric Weisstein's World of Mathematics, Rook Graph

T(n,k) = |A021010|. Sum of rows of T(n,k) is A002720. T(n,n) is the order of the symmetric group on an n-element set, n!.
Columns k=2..10 are A163102, A179058, A179059, A179060, A179061, A179062, A179063, A179064, A179065.
Cf. A089231, A105219, A295383.

/* As triangle */ [[(Binomial(n,k)^2)*Factorial(k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jun 13 2017

T:= (n, k)-> (binomial(n, k)^2)*k!:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 04 2012

Table[Table[Binomial[n, k]^2 k!,{k, 0, n}], {n, 0, 6}] // Flatten (* Geoffrey Critzer, Dec 04 2012 *)
Table[ CoefficientList[n!*LaguerreL[n, x], x] // Abs // Reverse, {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 18 2013 *)
CoefficientList[Table[n! x^n LaguerreL[n, -1/x], {n, 0, 8}], x] // Flatten (* Eric W. Weisstein, Apr 24 2017 *)
CoefficientList[Table[(-x)^n HypergeometricU[-n, 1, -(1/x)], {n, 5}],
  x] // Flatten (* Eric W. Weisstein, Jun 13 2017 *)

T(n,k) = k! * binomial(n,k)^2 \\ Andrew Howroyd, Feb 13 2018

T(n,k) = (C(n,k)^2)*k!.
T(n,k) = A007318(n,k) * A008279(n,k). - Antal Pinter, Nov 12 2014
From Peter Bala, Jul 04 2016: (Start)
G.f.: exp(x*t)*I_0(2*sqrt(x)) = 1 + (1 + t)*x/1!^2 + (1 + 4*t + 2*t^2)*x^2/2!^2 + (1 + 9*t + 18*t^2 + 6*t^3)*x^3/3!^2 + ..., where I_0(x) = Sum_{n >= 0} (x/2)^(2*n)/n!^2 is a modified Bessel function of the first kind.
The row polynomials R(n,t) satisfy R(n,t + u) = Sum_{k = 0..n} T(n,k)*t^k*R(n-k,u).
R(n,t) = 1 + Sum_{k = 0..n-1} (-1)^(n-k+1)*n!/k!*binomial(n,k) *t^(n-k)*R(k,t). Cf. A089231. (End)
From Peter Bala, Oct 05 2019: (Start)
E.g.f.: 1/(1 - t*x)*exp(x/(1 - t*x)).
Recurrence for row polynomials: R(n+1,t) = (1 + (2*n+1)*t)R(n,t) - n^2*t^2*R(n-1,t), with R(0,t) = 1 and R(1,t) = 1 + t.
R(n,t) equals the denominator polynomial of the finite continued fraction 1 + n*t/(1 + n*t/(1 + (n-1)*t/(1 + (n-1)*t/(1 + ... + 2*t/(1 + 2*t/(1 + t/(1 + t/(1)))))))). The numerator polynomial is the (n+1)-th row polynomial of A089231. (End)
Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/A001044(n) = exp(y*x)*E(x) where E(x) = Sum_{n>=0} x^n/A001044(n). - Geoffrey Critzer, Jan 08 2023
Sum_{k=0..n} k*T(n,k) = A105219(n). - Alois P. Heinz, Jan 08 2023
T(n,k) = Sum_{d=0..2*k} c(k,d)*n^d, where c(k,d) = Sum_{j=max(d-k,0)..k} binomial(k,j)*A008275(k+j,d)/j!. - Eder G. Santos, Jan 23 2025

A090582 T(n, k) = Sum_{j=0..n-k} (-1)^jbinomial(n - k + 1, j)(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 36, 14, 1, 120, 240, 150, 30, 1, 720, 1800, 1560, 540, 62, 1, 5040, 15120, 16800, 8400, 1806, 126, 1, 40320, 141120, 191520, 126000, 40824, 5796, 254, 1, 362880, 1451520, 2328480, 1905120, 834120, 186480, 18150, 510, 1, 3628800, 16329600, 30240000, 29635200, 16435440, 5103000, 818520, 55980, 1022, 1

Offset: 1

Hugo Pfoertner, Jan 11 2004

Let Q(m, n) = Sum_(k=0..n-1) (-1)^k * binomial(n, k) * (n-k)^m. Then Q(m,n) is the numerator of the probability P(m,n) = Q(m,n)/n^m of seeing each card at least once if m >= n cards are drawn with replacement from a deck of n cards, written in a two-dimensional array read by antidiagonals with Q(m,m) as first row and Q(m,1) as first column.
The sequence is given as a matrix with the first row containing the cases #draws = size_of_deck. The second row contains #draws = 1 + size_of_deck. If "mn" indicates m cards drawn from a deck with n cards then the locations in the matrix are:
  11 22 33 44 55 66 77 ...
  21 32 43 54 65 76 87 ...
  31 42 53 64 75 86 97 ...
  41 52 63 74 85 .. .. ...
read by antidiagonals ->:
  11, 22, 21, 33, 32, 31, 44, 43, 42, 41, 55, 54, 53, 52, ....
The probabilities are given by Q(m,n)/n^m:
.(m,n):.....11..22..21..33..32..31..44..43..42..41...55...54..53..52..51
.....Q:......1...2...1...6...6...1..24..36..14...1..120..240.150..30...1
...n^m:......1...4...1..27...8...1.256..81..16...1.3125.1024.243..32...1
%.Success:.100..50.100..22..75.100...9..44..88.100....4...23..62..94.100
P(n,n) = n!/n^n which can be approximated by sqrt(Pi*(2n+1/3))/e^n (Gosper's approximation to n!).
Let P[n] be the set of all n-permutations. Build a superset Q[n] of P[n] composed of n-permutations in which some (possibly all or none) ascents have been designated. An ascent in a permutation s[1]s[2]...s[n] is a pair of consecutive elements s[i],s[i+1] such that s[i] < s[i+1]. As a triangular array read by rows T(n,k) is the number of elements in Q[n] that have exactly k distinguished ascents, n >= 1, 0 <= k <= n-1. Row sums are A000670. E.g.f. is y/(1+y-exp(y*x)). For example, T(3,1)=6 because there are four 3-permutations with one ascent, with these we would also count 1->2 3, and 1 2->3 where exactly one ascent is designated by "->". (After Flajolet and Sedgewick.) - Geoffrey Critzer, Nov 13 2012
Sum_(k=1..n) Q(n, k)*binomial(x, k) = x^n such that Sum_{k=1..n} Q(n, i)*binomial(x+1,i+1) = Sum_{k=1..x} k^n. - David A. Corneth, Feb 17 2014
A141618(n,n-k+1) = a(n,k) * C(n,k-1) / k. - Tom Copeland, Oct 25 2014
See A074909 and above g.f.s below for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - Tom Copeland, Nov 14 2014
For connections to toric varieties and Eulerian polynomials (in addition to those noted below), see the Dolgachev and Lunts and the Stembridge links in A019538. - Tom Copeland, Dec 31 2015
See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra (this entry) and stellahedra. - Tom Copeland, Nov 14 2016
From the Hasan and Franco and Hasan papers: The m-permutohedra for m=1,2,3,4 are the line segment, hexagon, truncated octahedron and omnitruncated 5-cell. The first three are well-known from the study of elliptic models, brane tilings and brane brick models. The m+1 torus can be tiled by a single (m+2)-permutohedron. Relations to toric Calabi-Yau Kahler manifolds are also discussed. - Tom Copeland, May 14 2020

			For m = 4, n = 2, we draw 4 times from a deck of two cards. Call the cards "a" and "b" - of the 16 possible combinations of draws (each of which is equally likely to occur), only two do not contain both a and b: a, a, a, a and b, b, b, b. So the probability of seeing both a and b is 14/16. Therefore Q(m, n) = 14.
Table starts:
  [1] 1;
  [2] 2,      1;
  [3] 6,      6,       1;
  [4] 24,     36,      14,      1;
  [5] 120,    240,     150,     30,      1;
  [6] 720,    1800,    1560,    540,     62,     1;
  [7] 5040,   15120,   16800,   8400,    1806,   126,    1;
  [8] 40320,  141120,  191520,  126000,  40824,  5796,   254,   1;
  [9] 362880, 1451520, 2328480, 1905120, 834120, 186480, 18150, 510, 1.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Kevin Buhr, Michael Press and DMB, Collecting a deck of cards on the street. Thread in NG sci.math.
José L. Cereceda, Figurate numbers and sums of powers of integers, arXiv:2001.03208 [math.NT], 2020. See triangle Table 1 p. 5.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 209.
S. Franco and A. Hasan, Graded Quivers, Generalized Dimer Models and Toric Geometry, arXiv preprint arXiv:1904.07954 [hep-th], 2019.
A. Hasan, Physics and Mathematics of Graded Quivers, dissertation, Graduate Center, City University of New York, 2019.
Hugo Pfoertner, "Hit all" probabilities: Program and results.

Cf. A073593 first m >= n giving at least 50% probability, A085813 ditto for 95%, A055775 n^n/n!, A090583 Gosper's approximation to n!.
Reflected version of A019538.
Cf. A233734 (central terms).
Cf. A074909, A008292, A028246.
Cf. A046802, A119879, A123125, A130850, and A248727.

a090582 n k = a090582_tabl !! (n-1) !! (k-1)
a090582_row n = a090582_tabl !! (n-1)
a090582_tabl = map reverse a019538_tabl
-- Reinhard Zumkeller, Dec 15 2013

T := (n, k) -> add((-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n, j = 0..n-k):
# Or:
T := (n, k) -> (n - k + 1)!*Stirling2(n, n - k + 1):
for n from 1 to 9 do seq( T(n, k), k = 1..n) od; # Peter Luschny, May 21 2021

In[1]:= Table[Table[k! StirlingS2[n, k], {k, n, 1, -1}], {n, 1, 6}] (* Victor Adamchik, Oct 05 2005 *)
nn=6; a=y/(1+y-Exp[y x]); Range[0,nn]! CoefficientList[Series[a, {x,0,nn}], {x,y}]//Grid (* Geoffrey Critzer, Nov 10 2012 *)

a(n)={m=ceil((-1+sqrt(1+8*n))/2);k=m+1+binomial(m,2)-n;k*sum(i=1,k,(-1)^(i+k)*i^(m-1)*binomial(k-1,i-1))} \\ David A. Corneth, Feb 17 2014

T(n, k) = (n - k + 1)!*Stirling2(n, n - k + 1), generated by Stirling numbers of the second kind multiplied by a factorial. - Victor Adamchik, Oct 05 2005
Triangle T(n,k), 1 <= k <= n, read by rows given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] DELTA [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2006
From Tom Copeland, Oct 07 2008: (Start)
G(x,t) = 1/ (1 + (1-exp(x*t))/t) = 1 + 1*x + (2 + t)*x^2/2! + (6 + 6*t + t^2)*x^3/3! + ... gives row polynomials of A090582, the f-polynomials for the permutohedra (see A019538).
G(x,t-1) = 1 + 1*x + (1 + t)*x^2/2! + (1 + 4*t + t^2)*x^3/3! + ... gives row polynomials for A008292, the h-polynomials of permutohedra.
G[(t+1)x,-1/(t+1)] = 1 + (1 + t)*x + (1 + 3*t + 2*t^2)*x^2/2! + ... gives row polynomials of A028246. (End)
From Tom Copeland, Oct 11 2011: (Start)
With e.g.f. A(x,t) = G(x,t) - 1, the compositional inverse in x is
B(x,t) = log((t+1)-t/(1+x))/t. Let h(x,t) = 1/(dB/dx) = (1+x)*(1+(1+t)x), then the row polynomial P(n,t) is given by (1/n!)*(h(x,t)*d/dx)^n x, evaluated at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). (End)
k <= 0 or k > n yields Q(n, k) = 0; Q(1,1) = 1; Q(n, k) = k * (Q(n-1, k) + Q(n-1, k-1)). - David A. Corneth, Feb 17 2014
T = A008292*A007318. - Tom Copeland, Nov 13 2016
With all offsets 0 for this entry, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125 with offsets -1 so that the array becomes A008292; i.e., we ignore the first row and first column of A123125. Then the row polynomials of this entry, A090582, are given by A_n(1 + x;0). Other specializations of A_n(x;y) give A028246, A046802, A119879, A130850, and A248727. - Tom Copeland, Jan 24 2020

A229223 Number G(n,k) of set partitions of {1,...,n} into sets of size at most k; triangle G(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 10, 14, 15, 0, 1, 26, 46, 51, 52, 0, 1, 76, 166, 196, 202, 203, 0, 1, 232, 652, 827, 869, 876, 877, 0, 1, 764, 2780, 3795, 4075, 4131, 4139, 4140, 0, 1, 2620, 12644, 18755, 20645, 21065, 21137, 21146, 21147

Offset: 0

Alois P. Heinz, Sep 16 2013

John Riordan calls these Allied Bell Numbers. - N. J. A. Sloane, Jan 10 2018
G(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. G(n,k) = G(n,n) = A000110(n) for k>n.
G(n,k) - G(n,k-1) = A080510(n,k).
A column G(n>=0,k) can be generated by a linear recurrence with polynomial coefficients, where the initial terms correspond with A000110, and the coefficients contain constant factors derived from A008279 (cf. recg(k) in the fourth Maple program below). - Georg Fischer, May 19 2021

			G(4,2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.
Triangle G(n,k) begins:
  1;
  0,  1;
  0,  1,   2;
  0,  1,   4,    5;
  0,  1,  10,   14,   15,
  0,  1,  26,   46,   51,   52;
  0,  1,  76,  166,  196,  202,  203;
  0,  1, 232,  652,  827,  869,  876,  877;
  0,  1, 764, 2780, 3795, 4075, 4131, 4139, 4140;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
J. Riordan, Letter, 11/23/1970

Columns k=0-10 give: A000007, A000012, A000085, A001680, A001681, A110038, A148092, A229224, A229225, A229226, A229227.
Main diagonal gives: A000110. Lower diagonal gives: A058692.
Cf. A066223 (G(2n,2)), A229228 (G(2n,n)), A229229 (G(n^2,n)), A227223 (G(2^n,n)).
Cf. A008279, A080510.

G:= proc(n, k) option remember; `if`(n=0, 1, `if`(k<1, 0,
       add(G(n-k*j, k-1) *n!/k!^j/(n-k*j)!/j!, j=0..n/k)))
    end:
seq(seq(G(n, k), k=0..n), n=0..10);
# second Maple program:
G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j,k) od; % fi
    end:
seq(seq(G(n, k), k=0..n), n=0..10);
# third Maple program:
G:= proc(n, k) option remember; `if`(n=0, 1, add(
      G(n-i, k)*binomial(n-1, i-1), i=1..min(n, k)))
    end:
seq(seq(G(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 26 2017
# fourth Maple program (for columns G(n>=0,k)):
init := n -> seq(a(j) = combinat:-bell(j), j=0..n): # A000110
b := (n, k) -> mul((n - j)/(j + 1), j = 0..k-1):
recg := k -> {(k-1)!*(add(j*b(n, j)*a(n-j), j = 1..k) - n*a(n)), init(k-1)}:
column := proc(k, len) local f; f := gfun:-rectoproc(recg(k), a(n), remember):
map(f, [$0..len-1]) end:
seq(print(column(k, 12)), k=1..9); # Georg Fischer, May 19 2021

g[n_, k_] := g[n, k] = If[n == 0, 1, If[k < 1, 0, Sum[g[n - k*j, k - 1] *n!/k!^j/(n - k*j)!/j!, { j, 0, n/k}]]]; Table[Table[g[n, k], { k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

G(0,k) = 1, G(n,k) = 0 for n>0 and k<1, otherwise G(n,k) = Sum_{j=0..floor(n/k)} G(n-k*j,k-1) * n!/(k!^j*(n-k*j)!*j!).
G(n,k) = G(n-1,k) +(n-1)/1 *(G(n-2,k) +(n-2)/2 *(G(n-3,k) +(n-3)/3 *(G(n-4,k) + ... +(n-(k-1))/(k-1) *G(n-k,k)...))).
E.g.f. of column k: exp(Sum_{j=1..k} x^j/j!).

A009963 Triangle of numbers n!(n-1)!...(n-k+1)!/(1!2!...k!).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 24, 72, 24, 1, 1, 120, 1440, 1440, 120, 1, 1, 720, 43200, 172800, 43200, 720, 1, 1, 5040, 1814400, 36288000, 36288000, 1814400, 5040, 1, 1, 40320, 101606400, 12192768000, 60963840000, 12192768000, 101606400, 40320, 1

Offset: 0

N. J. A. Sloane

Product of all matrix elements of n X k matrix M(i,j) = i+j (i=1..n-k, j=1..k). - Peter Luschny, Nov 26 2012

These are the generalized binomial coefficients associated to the sequence A000178. - Tom Edgar, Feb 13 2014

			Rows start:
  1;
  1,   1;
  1,   2,    1;
  1,   6,    6,    1;
  1,  24,   72,   24,   1;
  1, 120, 1440, 1440, 120, 1;  etc.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000178, A007318, A060854, A090441.
Central column is A079478.
Columns include A010796, A010797, A010798, A010799, A010800.
Row sums give A193520.

A009963:= func< n,k | (1/Factorial(n+1))*(&*[ Factorial(n-j+1)/Factorial(j): j in [0..k]]) >;
[A009963(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 04 2022

(* First program *)
row[n_]:= Table[Product[i+j, {i,1,n-k}, {j,1,k}], {k,0,n}];
Array[row, 9, 0] // Flatten (* Jean-François Alcover, Jun 01 2019, after Peter Luschny *)
(* Second program *)
T[n_, k_]:= BarnesG[n+2]/(BarnesG[k+2]*BarnesG[n-k+2]);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 04 2022 *)

def A009963_row(n):
    return [mul(mul(i+j for j in (1..k)) for i in (1..n-k)) for k in (0..n)]
for n in (0..7): A009963_row(n)  # Peter Luschny, Nov 26 2012

def triangle_to_n_rows(n): #changing n will give you the triangle to row n.
    N=[[1]+n*[0]]
    for i in [1..n]:
        N.append([])
        for j in [0..n]:
            if i>=j:
                N[i].append(factorial(i-j)*binomial(i-1,j-1)*N[i-1][j-1]+factorial(j)*binomial(i-1,j)*N[i-1][j])
            else:
                N[i].append(0)
    return [[N[i][j] for j in [0..i]] for i in [0..n]]
    # Tom Edgar, Feb 13 2014

T(n,k) = T(n-1,k-1)*A008279(n,n-k) = A000178(n)/(A000178(k)*A000178(n-k)) i.e., a "supercombination" of "superfactorials". - Henry Bottomley, May 22 2002
Equals ConvOffsStoT transform of the factorials starting (1, 2, 6, 24, ...); e.g., ConvOffs transform of (1, 2, 6, 24) = (1, 24, 72, 24, 1). Note that A090441 = ConvOffsStoT transform of the factorials, A000142. - Gary W. Adamson, Apr 21 2008
Asymptotic: T(n,k) ~ exp((3/2)*k^2 - zeta'(-1) + 3/4 - (3/2)*n*k)*(1+n)^((1/2)*n^2 + n + 5/12)*(1+k)^(-(1/2)*k^2 - k - 5/12)*(1 + n - k)^(-(1/2)*n^2 + n*k - (1/2)*k^2 - n + k - 5/12)/(sqrt(2*Pi). - Peter Luschny, Nov 26 2012
T(n,k) = (n-k)!*C(n-1,k-1)*T(n-1,k-1) + k!*C(n-1,k)*T(n-1,k) where C(i,j) is given by A007318. - Tom Edgar, Feb 13 2014
T(n,k) = Product_{i=1..k} (n+1-i)!/i!. - Alois P. Heinz, Jun 07 2017
T(n,k) = BarnesG(n+2)/(BarnesG(k+2)*BarnesG(n-k+2)). - G. C. Greubel, Jan 04 2022

A078740 Triangle of generalized Stirling numbers S_{3,2}(n,k) read by rows (n>=1, 2<=k<=2n).

Original entry on oeis.org

1, 6, 6, 1, 72, 168, 96, 18, 1, 1440, 5760, 6120, 2520, 456, 36, 1, 43200, 259200, 424800, 285120, 92520, 15600, 1380, 60, 1, 1814400, 15120000, 34776000, 33566400, 16304400, 4379760, 682200, 62400, 3270, 90, 1, 101606400, 1117670400

Offset: 1

N. J. A. Sloane, Dec 21 2002

The sequence of row lengths of this array is [1,3,5,7,...] = A005408(n-1), n>=1.

For the scaled array s2_{3,2}(n,k) := a(n,k)*k!/((n+1)!*n!) see A090452.

			1;
6, 6, 1;
72, 168, 96, 18, 1;
...

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (rows 1 <= n <= 100, flattened).
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
W. Lang, First 6 rows.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Row sums give A078738. Cf. A078739.

Cf. A005408, A090452.

a[n_, k_] := (-1)^k*n!*(n+1)!*HypergeometricPFQ[{2-k, n+1, n+2}, {2, 3}, 1]/(2*(k-2)!); Table[a[n, k], {n, 1, 7}, {k, 2, 2*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

Recursion: a(n, k) = Sum(binomial(2, p)*fallfac(n-1-p+k, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 1)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=3, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
a(n, k) = (((-1)^k)/k!)*Sum(((-1)^p)*binomial(k, p)*product(fallfac(p+(j-1)*(3-2), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=3, s=2.
a(n, k) = (-1)^k n! (n+1)! 3F2(2-k, n+1, n+2; 2, 3; 1) / (2(k-2)!). - Jean-François Alcover, Dec 04 2013

Edited by Wolfdieter Lang, Dec 23 2003

A078741 Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).

Original entry on oeis.org

1, 6, 18, 9, 1, 36, 540, 1242, 882, 243, 27, 1, 216, 13608, 94284, 186876, 149580, 56808, 11025, 1107, 54, 1, 1296, 330480, 6148872, 28245672, 49658508, 41392620, 18428400, 4691412, 706833, 63375, 3285, 90, 1, 7776, 7954848, 380841264, 3762380016, 13062960720

Offset: 1

N. J. A. Sloane, Dec 21 2002

The sequence of row lengths for this array is [1,4,7,10,..]= A016777(n-1), n>=1.
The g.f. for the k-th column, (with leading zeros and k>=3) is G(k,x)= x^ceiling(k/3)*P(k,x)/product(1-fallfac(p,3)*x,p=3..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A089517(k,m)*x^m,m=0..kmax(k)), k>=3, with kmax(k) := A004523(k-3)= floor(2*(k-3)/3)= [0,0,1,2,2,3,4,4,5,...]. For the recurrence of the G(k,x) see A089517. Wolfdieter Lang, Dec 01 2003
For the computation of the k-th column sequence see A090219.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_3 (the disjoint union of n copies of the complete graph K_3). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k |   3     4     5      6      7     8     9   10  11  12
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
  1 |   1
  2 |   6    18     9      1
  3 |  36   540  1242    882    243    27     1
  4 | 216 13608 94284 186876 149580 56808 11025 1107  54   1
...
Graph coloring interpretation of T(2,3) = 6:
The graph 2K_3 is 2 copies of K_3, the complete graph on 3 vertices:
    o b      o e
   / \      / \
  o---o    o---o
  a   c    d   f
The six 3-colorings of 2K_3 are ad|be|cf, ad|bf|ce, ae|bd|cf, ae|bf|cd, af|bd|ce, and af|be|cd. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027,2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Codara, O. M. D’Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers arXiv:1308.1700v1 [cs.DM]
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
W. Lang, First 6 rows.

Row sums give A069223. Cf. A078739.
The column sequences (without leading zeros) are A000400 (powers of 6), 18*A089507, 9*A089518, A089519, etc.
A089504, A069223 (row sums), A090212 (alternating row sums).
A078740, A090214.

a[n_, k_] := (-1)^k*Sum[(-1)^p*((p-2)*(p-1)*p)^n*Binomial[k, p], {p, 3, k}]/k!; Table[a[n, k], {n, 1, 6}, {k, 3, 3*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

a(n, k) = (((-1)^k)/k!)*Sum_{p = 3..k} (-1)^p* binomial(k, p)*fallfac(p, 3)^n, with fallfac(p, 3) := A008279(p, 3) = p*(p-1)*(p-2); 3 <= k <= 3*n, n >= 1, else 0. From eq.(19) with r = 3 of the Blasiak et al. reference.
E^n = Sum_{k = 3..3*n} a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^3d^3/dx^3.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k >= 0} (k*(k-1)*(k-2))^n*x^k/k!. - Peter Bala, Aug 15 2013

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

A090438 Generalized Stirling2 array (4,2).

Original entry on oeis.org

1, 12, 8, 1, 360, 480, 180, 24, 1, 20160, 40320, 25200, 6720, 840, 48, 1, 1814400, 4838400, 4233600, 1693440, 352800, 40320, 2520, 80, 1, 239500800, 798336000, 898128000, 479001600, 139708800, 23950080, 2494800, 158400, 5940, 120, 1

Offset: 1

Wolfdieter Lang, Dec 23 2003

The row length sequences for this array is [1,3,5,7,9,11,...] = A005408(n-1), n>=1.

The scaled array a(n,k)/((2*n)!/k!) = A034870(n-1,k-2), n>=1, 2<=k<=2*n (Pascal triangle, even numbered rows only).

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
Wolfdieter Lang, First 6 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A078740 (3, 2)-Stirling2.
Cf. A072678 (row sums), A090439 (alternating row sums).
Cf. A062139.

with(PolynomialTools):
p := n -> (2*n+2)!*hypergeom([-2*n],[3], -x)/2:
seq(CoefficientList(simplify(p(n)),x), n=0..5); # Peter Luschny, Apr 08 2015

a[n_, k_] := (-1)^k/k!*Sum[(-1)^p*Binomial[k, p]*Product[FactorialPower[p + 2*(j-1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2 n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

Recursion: a(n, k) = sum(binomial(2, p)*fallfac(2*(n-1)-p+k, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=4, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+2*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=4, s=2.
a(n, k) = ((2*n)!/k!)*binomial(2*(n-1), k-2), n>=1, 2<=k<=2*n.
E.g.f. column k>=2 (with leading zeros): (((-1)^k)/k!)*(sum(((-1)^p)*binomial(k, p)*hypergeom([(p-1)/2, p/2], [], 4*x), p=2..k)-(k-1)).
Coefficient triangle of the polynomials (2*n+2)!*hypergeom([-2*n],[3],-x)/2. - Peter Luschny, Apr 08 2015
Coefficient triangle of Laguerre polynomials (2*n)!*L(2*n,2,-x). - Peter Luschny, Apr 08 2015

A003470 a(n) = n*a(n-1) - a(n-2) + 1 + (-1)^n.

Original entry on oeis.org

1, 1, 3, 8, 31, 147, 853, 5824, 45741, 405845, 4012711, 43733976, 520795003, 6726601063, 93651619881, 1398047697152, 22275111534553, 377278848390249, 6768744159489931, 128228860181918440, 2557808459478878871, 53585748788874537851, 1176328664895760953853

Offset: 0

N. J. A. Sloane and John Riordan

Row sums of A086764. - Philippe Deléham, Apr 27 2004

a(n+2m) == a(n) (mod m) for all n and m. - Robert Israel, Dec 06 2016

			G.f. = 1 + x + 3*x^2 + 8*x^3 + 31*x^4 + 147*x^5 + 853*x^6 + 5824*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
John Riordan and N. J. A. Sloane, Correspondence, 1974

Cf. A008279, A072374, A086764, A143409.

f:= gfun:-rectoproc({a(n) -(n-1)*a(n-1)-(n-2)*a(n-2)+a(n-3)-2=0,a(0)=1,a(1)=1,a(2)=3},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 06 2016

t = {1, 1}; Do[AppendTo[t, n*t[[-1]] - t[[-2]] + 1 + (-1)^n], {n, 2, 20}] (* T. D. Noe, Oct 07 2013 *)
T[n_, k_] := HypergeometricPFQ[{k+1, k-n},{},-1];
Table[Sum[(-1)^k T[n,k], {k,0,n}], {n,0,22}] (* Peter Luschny, Oct 05 2017 *)

Diagonal sums of reverse of permutation triangle A008279. a(n) = Sum_{k=0..floor(n/2)} (n-k)!/k!. - Paul Barry, May 12 2004
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(n-2k)!. - Paul Barry Dec 15 2010
G.f.: 1/(1-x^2-x/(1-x/(1-x^2-2x/(1-2x/(1-x^2-3x/(1-3x/(1-x^2-4x/(1-4x/(1-.... (continued fraction);
G.f.: 1/(1-x-x^2-x^2/(1-3x-x^2-4x^2/(1-5x-x^2-9x^2/(1-7x-x^2-16x^2/(1-... (continued fraction). - Paul Barry, Dec 15 2010
G.f.: hypergeom([1,1],[],x/(1-x^2))/(1-x^2). - Mark van Hoeij, Nov 08 2011
G.f.: 1/Q(0), where Q(k)= 1 - x^2 - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013
From Robert Israel, Dec 06 2016: (Start)
a(2m) = hypergeom([1,-m,m+1],[],-1).
a(2m+1) = hypergeom([1,-m,m+2],[],-1)*(m+1).
a(2m-1) + a(2m+1) = (2m+1) a(2m). (End)
0 = a(n)*(-a(n+2) - a(n+3)) + a(n+1)*(-2 + a(n+1) - 2*a(n+3) + a(n+4)) + a(n+2)*(-2*a(n+3) + a(n+4)) + a(n+3)*(+2 - a(n+3)) if n >= 0. - Michael Somos, Dec 06 2016
0 = a(n)*(-a(n+2) + a(n+4)) + a(n+1)*(+a(n+1) - a(n+2) - a(n+3) + 3*a(n+4) - a(n+5)) + a(n+2)*(-a(n+3) + a(n+4)) + a(n+3)*(-a(n+4) + a(n+5)) + a(n+4)*(-a(n+4)) if n >= 0. - Michael Somos, Dec 06 2016
a(n) = Sum_{k=0..n} (-1)^k*hypergeom([k+1, k-n], [], -1). - Peter Luschny, Oct 05 2017
D-finite with recurrence: a(n) -n*a(n-1) +(n-2)*a(n-3) -a(n-4)=0. - R. J. Mathar, Apr 29 2020
a(n) ~ n! * (1 + 1/n + 1/(2*n^2) + 2/(3*n^3) + 25/(24*n^4) + 77/(40*n^5) + 2971/(720*n^6) + 6287/(630*n^7) + 1074809/(40320*n^8) + 28160749/(362880*n^9) + ...). - Vaclav Kotesovec, Nov 25 2022

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 25 2004

cp's OEIS Frontend

A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

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A144084 T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

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A090582 T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.

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A229223 Number G(n,k) of set partitions of {1,...,n} into sets of size at most k; triangle G(n,k), n>=0, 0<=k<=n, read by rows.

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A009963 Triangle of numbers n!(n-1)!...(n-k+1)!/(1!2!...k!).

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A078740 Triangle of generalized Stirling numbers S_{3,2}(n,k) read by rows (n>=1, 2<=k<=2n).

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A090582 T(n, k) = Sum_{j=0..n-k} (-1)^jbinomial(n - k + 1, j)(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.