A025211 -id:A025211 - cp's OEIS frontend

A143494 Triangle read by rows: 2-Stirling numbers of the second kind.

1, 2, 1, 4, 5, 1, 8, 19, 9, 1, 16, 65, 55, 14, 1, 32, 211, 285, 125, 20, 1, 64, 665, 1351, 910, 245, 27, 1, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1, 512, 19171, 111645, 204205, 156660, 58107, 11130, 1110, 54, 1

Offset: 2

Peter Bala, Aug 20 2008

This is the case r = 2 of the r-Stirling numbers of the second kind. The 2-Stirling numbers of the second kind give the number of ways of partitioning the set {1,2,...,n} into k nonempty disjoint subsets with the restriction that the elements 1 and 2 belong to distinct subsets.
More generally, the r-Stirling numbers of the second kind give the number of ways of partitioning the set {1,2,...,n} into k nonempty disjoint subsets with the restriction that the numbers 1, 2, ..., r belong to distinct subsets. The case r = 1 gives the usual Stirling numbers of the second kind A008277; for other cases see A143495 (r = 3) and A143496 (r = 4).
The lower unitriangular array of r-Stirling numbers of the second kind equals the matrix product P^(r-1) * S (with suitable offsets in the row and column indexing), where P is Pascal's triangle, A007318 and S is the array of Stirling numbers of the second kind, A008277.
For the definition of and entries relating to the corresponding r-Stirling numbers of the first kind see A143491. For entries on r-Lah numbers refer to A143497. The theory of r-Stirling numbers of both kinds is developed in [Broder].
From Peter Bala, Sep 19 2008: (Start)
Let D be the derivative operator d/dx and E the Euler operator x*d/dx. Then x^(-2)*E^n*x^2 = Sum_{k = 0..n} T(n+2,k+2)*x^k*D^k.
The row generating polynomials R_n(x) := Sum_{k= 2..n} T(n,k)*x^k satisfy the recurrence R_(n+1)(x) = x*R_n(x) + x*d/dx(R_n(x)) with R_2(x) = x^2. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).
Relation with the 2-Eulerian numbers E_2(n,j) := A144696(n,j): T(n,k) = 2!/k!*Sum_ {j = n-k..n-2} E_2(n,j)*binomial(j,n-k) for n >= k >= 2. (End)
From Wolfdieter Lang, Sep 29 2011: (Start)
T(n,k) = S(n,k,2), n>=k>=2, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column no. k from (A20) with k->2, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(2*x),exp(x)-1) with e.g.f. of column no. m>=0: exp(2*x)*((exp(x)-1)^m)/m!. See one of the formulas given below. For Sheffer matrices see the W. Lang link under A006232 with the S. Roman reference, also found in A132393. (End)

			Triangle begins
  n\k|...2....3....4....5....6....7
  =================================
  2..|...1
  3..|...2....1
  4..|...4....5....1
  5..|...8...19....9....1
  6..|..16...65...55...14....1
  7..|..32..211..285..125...20....1
  ...
T(4,3) = 5. The set {1,2,3,4} can be partitioned into three subsets such that 1 and 2 belong to different subsets in 5 ways: {{1}{2}{3,4}}, {{1}{3}{2,4}}, {{1}{4}{2,3}}, {{2}{3}{1,4}} and {{2}{4}{1,3}}; the remaining possibility {{1,2}{3}{4}} is not allowed.
From _Peter Bala_, Feb 23 2025: (Start)
The array factorizes as
/ 1               \       /1             \ /1             \ /1            \
| 2    1           |     | 2   1          ||0  1           ||0  1          |
| 4    5   1       |  =  | 4   3   1      ||0  2   1       ||0  0  1       | ...
| 8   19   9   1   |     | 8   7   4  1   ||0  4   3  1    ||0  0  2  1    |
|16   65  55  14  1|     |16  15  11  6  1||0  8   7  4  1 ||0  0  4  3  1 |
|...               |     |...             ||...            ||...           |
where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 2*x), x/(1 - x)). See A055248. (End)

Peter Bala, Factorising (r,b)-Stirling arrays
S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 2.
Andrei Z. Broder, The r-Stirling numbers, Report Number: CS-TR-82-949, 1982, Stanford University, Department of Computer Science.
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
C. B. Corcino, L. C. Hsu, and E. L. Tan, Asymptotic approximations of r-Stirling numbers, Approximation Theory Appl. 15, No. 3 13-25 (1999).
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764 [math.CO], 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.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
V. V. Mikhailov, Ordering of some boson operator functions, J. Phys A: Math. Gen. 16 (1983) 3817-3827.
V. V. Mikhailov, Normal ordering and generalised Stirling numbers, J. Phys A: Math. Gen. 18 (1985) 231-235.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Technical Report TR 99-05, July 1999, Universität Wien.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).
M. d’Ocagne, Sur une classe de nombres remarquables, Amer. J. Math., Vol. 9 (1887), 353-380.
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.
Mark Shattuck, Generalized r-Lah numbers, arXiv:1412.8721 [math.CO], 2014.

A001047 (column 3), A005493 (row sums), A008277, A016269 (column 4), A025211 (column 5), A049444 (matrix inverse), A074051 (alt. row sums).

Cf. A055248, A143491, A143495, A143496, A143497.

with combinat: T := (n, k) -> (1/(k-2)!)*add ((-1)^(k-i)*binomial(k-2,i)*(i+2)^(n-2),i = 0..k-2): for n from 2 to 11 do seq(T(n, k), k = 2..n) end do;

t[n_, k_] := StirlingS2[n, k] - StirlingS2[n-1, k]; Flatten[ Table[ t[n, k], {n, 2, 11}, {k, 2, n}]] (* Jean-François Alcover, Dec 02 2011 *)

@CachedFunction
def stirling2r(n, k, r) :
    if n < r: return 0
    if n == r: return 1 if k == r else 0
    return stirling2r(n-1,k-1,r) + k*stirling2r(n-1,k,r)
A143494 = lambda n,k: stirling2r(n, k, 2)
for n in (2..6):
    [A143494(n, k) for k in (2..n)] # Peter Luschny, Nov 19 2012

T(n+2,k+2) = (1/k!)*Sum_{i = 0..k} (-1)^(k-i)*C(k,i)*(i+2)^n, n,k >= 0.
T(n,k) = Stirling2(n,k) - Stirling2(n-1,k) for n, k >= 2.
Recurrence relation: T(n,k) = T(n-1,k-1) + k*T(n-1,k) for n > 2, with boundary conditions T(n,1) = T(1,n) = 0 for all n, T(2,2) = 1 and T(2,k) = 0 for k > 2. Special cases: T(n,2) = 2^(n-2); T(n,3) = 3^(n-2) - 2^(n-2).
As a sum of monomial functions of degree m: T(n+m,n) = Sum_{2 <= i_1 <= ... <= i_m <= n} (i_1*i_2*...*i_m). For example, T(6,4) = Sum_{2 <= i <= j <= 4} (i*j) = 2*2 + 2*3 + 2*4 + 3*3 + 3*4 + 4*4 = 55.
E.g.f. column k+2 (with offset 2): 1/k!*exp(2*x)*(exp(x) - 1)^k.
O.g.f. k-th column: Sum_{n >= k} T(n,k)*x^n = x^k/((1-2*x)*(1-3*x)*...*(1-k*x)).
E.g.f.: exp(2*t + x*(exp(t) - 1)) = Sum_{n >= 0} Sum_{k = 0..n} T(n+2,k+2) *x^k*t^n/n! = Sum_{n >= 0} B_n(2;x)*t^n/n! = 1 + (2 + x)*t/1! + (4 + 5*x + x^2)*t^2/2! + ..., where the row polynomial B_n(2;x) := Sum_{k = 0..n} T(n+2,k+2)*x^k denotes the 2-Bell polynomial.
Dobinski-type identities: Row polynomial B_n(2;x) = exp(-x)*Sum_{i >= 0} (i + 2)^n*x^i/i!. Sum_{k = 0..n} k!*T(n+2,k+2)*x^k = Sum_{i >= 0} (i + 2)^n*x^i/(1 + x)^(i+1).
The T(n,k) are the connection coefficients between falling factorials and the shifted monomials (x + 2)^(n-2). For example, from row 4 we have 4 + 5*x + x*(x - 1) = (x + 2)^2, while from row 5 we have 8 + 19*x + 9*x*(x - 1) + x*(x - 1)*(x - 2) = (x + 2)^3.
The row sums of the array are the 2-Bell numbers, B_n(2;1), equal to A005493(n-2). The alternating row sums are the complementary 2-Bell numbers, B_n(2;-1), equal to (-1)^n*A074051(n-2).
This array is the matrix product P * S, where P denotes the Pascal triangle, A007318 and S denotes the lower triangular array of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]).
Also, this array equals the transpose of the upper triangular array A126351. The inverse array is A049444, the signed 2-Stirling numbers of the first kind. See A143491 for the unsigned version of the inverse.
Let f(x) = exp(exp(x)). Then for n >= 1, the row polynomials R(n,x) are given by R(n+2,exp(x)) = 1/f(x)*(d/dx)^n(exp(2*x)*f(x)). Similar formulas hold for A008277, A039755, A105794, A111577 and A154537. - Peter Bala, Mar 01 2012

A028025 Expansion of 1/((1-3x)(1-4x)(1-5x)*(1-6x)).

Original entry on oeis.org

1, 18, 205, 1890, 15421, 116298, 830845, 5709330, 38119741, 249026778, 1599719485, 10142356770, 63639854461, 396031348458, 2448208592125, 15053605980210, 92160458747581, 562225198873338, 3419937140824765

Offset: 0

N. J. A. Sloane

This gives the fourth column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - Wolfdieter Lang, Oct 08 2011

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-119,342,-360).

Cf. A000453, A025211, A003468, A028165, A028200, A016109, A016075, A016094.

Cf. A143495, A193685.

CoefficientList[Series[1/((1-3x)(1-4x)(1-5x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{18,-119,342,-360},{1,18,205,1890},30] (* Harvey P. Dale, Jan 29 2024 *)

Vec(1/((1-3*x)*(1-4*x)*(1-5*x)*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,3), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = -5^(n+3)/2 + 2*4^(n+2)+ 6^(n+2) - 3^(n+2)/2. - R. J. Mathar, Mar 22 2011
O.g.f.: 1/((1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)).
E.g.f.: (d^3/dx^3)(exp(3*x)*((exp(x)-1)^3)/3!). - Wolfdieter Lang, Oct 08 2011

A016075 Expansion of 1/((1-8x)(1-9x)(1-10x)(1-11*x)).

Original entry on oeis.org

1, 38, 905, 17290, 289821, 4453638, 64331905, 887339330, 11810819141, 152832918238, 1933092302505, 23997027406170, 293289532268461, 3537885908902838, 42204462297434705, 498697803478957810, 5844588402226277781, 68011678300853991438, 786547256602640400505

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (38,-539,3382,-7920)

Cf. A000453, A025211, A028025, A003468, A028165, A028200, A016109, A016094.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-8*x)*(1-9*x)*(1-10*x)*(1-11*x)))); // Vincenzo Librandi, Jun 24 2013

I:=[1, 38, 905, 17290]; [n le 4 select I[n] else 38*Self(n-1)-539*Self(n-2)+3382*Self(n-3)-7920*Self(n-4): n in [1..20]]; // Vincenzo Librandi, Jun 24 2013

CoefficientList[Series[1/((1-8*x)*(1-9*x)*(1-10*x)*(1-11*x)), {x,0,20}], x] (* Vincenzo Librandi, Jun 23 2013 *)

x='x+O('x^30); Vec(1/((1-8*x)*(1-9*x)*(1-10*x)*(1-11*x))) \\ G. C. Greubel, Feb 07 2018

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,8), (n>=3). - Milan Janjic, Apr 26 2009
a(n) = 38*a(n-1) - 539*a(n-2) + 3382*a(n-3) - 7920*a(n-4), n>=4. - Vincenzo Librandi, Mar 17 2011
a(n) = 21*a(n-1) - 110*a(n-2) + 9^(n+1) - 8^(n+1), n>=2. - Vincenzo Librandi, Mar 17 2011
a(n) = 11^(n+3)/6 -5*10^(n+2) -4*8^(n+2)/3 + 9^(n+3)/2. - R. J. Mathar, Mar 18 2011

A028165 Expansion of 1/((1-5x)(1-6x)(1-7x)*(1-8x)).

Original entry on oeis.org

1, 26, 425, 5590, 64701, 688506, 6906145, 66324830, 616252901, 5580303586, 49508360265, 432061044870, 3720287489101, 31681154472266, 267320885100785, 2238337148081710, 18621251375573301, 154069635600426546

Offset: 0

N. J. A. Sloane

This is the column m=2 sequence (without leading zeros) of the Sheffer triangle (exp(5*x), exp(x)-1) of the 5-restricted Stirling2 numbers A193685. For a proof see the column o.g.f. formula there. - Wolfdieter Lang, Oct 07 2011

Index entries for linear recurrences with constant coefficients, signature (26, -251, 1066, -1680).

Cf. A000351, A005062, A019757, A193685.

Cf. A000453, A025211, A028025, A003468, A028200, A016109, A016075, A016094.

Vec(1/((1-5*x)*(1-6*x)*(1-7*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,5), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = 26*a(n-1) - 251*a(n-2) + 1066*a(n-3) - 1680*a(n-4), n >= 4. - Vincenzo Librandi, Mar 19 2011
a(n) = 15*a(n-1) - 56*a(n-2) + 6^(n+1) - 5^(n+1), a(0)=1, a(1)=26. - Vincenzo Librandi, Mar 19 2011
E.g.f.: (d^3/dx^3)(exp(5*x)*((exp(x)-1)^3)/3!). See the Sheffer triangle comment above. - Wolfdieter Lang, Oct 07 2011
a(n) = -125*5^n/6 + 108*6^n - 343*7^n/2 + 256*8^n/3. - R. J. Mathar, Jun 23 2013

A028200 Expansion of 1/((1-6x)(1-7x)(1-8x)*(1-9x)).

Original entry on oeis.org

1, 30, 565, 8550, 113701, 1388310, 15958405, 175419750, 1863406501, 19269697590, 195034120645, 1939826329350, 19018419228901, 184245490086870, 1767124523521285, 16805853434269350, 158682246543588901, 1489103597614860150, 13900428943759584325

Offset: 0

N. J. A. Sloane

nonn

Matthew House, Table of n, a(n) for n = 0..1040
Index entries for linear recurrences with constant coefficients, signature (30,-335,1650,-3024).

Cf. A000453, A025211, A028025, A003468, A028165, A016109, A016075, A016094.

CoefficientList[Series[ 1/((1-6x)(1-7x)(1-8x)(1-9x)), {x, 0, 20} ], x]
LinearRecurrence[{30,-335,1650,-3024},{1,30,565,8550},20] (* Harvey P. Dale, Mar 27 2023 *)

Vec(1/((1-6*x)*(1-7*x)*(1-8*x)*(1-9*x)) + O(x^30)) \\ Michel Marcus, Feb 12 2017

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,6), (n >= 3). [Milan Janjic, Apr 26 2009]
a(n) = 17*a(n-1) - 72*a(n-2) + 7^(n+1) - 6^(n+1), a(0)=1, a(1)=30. - Vincenzo Librandi, Mar 11 2011
a(n) = (9^(n+3) - 3*8^(n+3) + 3*7^(n+3) - 6^(n+3))/6. [Yahia Kahloune, Jun 12 2013]
a(n) = 30*a(n-1) - 335*a(n-2) + 1650*a(n-3) - 3024*a(n-4). - Matthew House, Feb 11 2017

A016094 Expansion of 1/((1-9x)(1-10x)(1-11x)(1-12*x)).

Original entry on oeis.org

1, 42, 1105, 23310, 431221, 7309722, 116419465, 1769717670, 25948716541, 369730963602, 5147200519825, 70298695224030, 944897655707461, 12530341519244682, 164265473257148185, 2132247784185258390

Offset: 0

N. J. A. Sloane

nonn

Index entries for linear recurrences with constant coefficients, signature (42,-659,4578,-11880)

Cf. A000453, A025211, A028025, A003468, A028165, A028200, A016109, A016075.

CoefficientList[Series[1/((1-9x)(1-10x)(1-11x)(1-12x)) ,{x,0,20}],x] (* or *) LinearRecurrence[{42,-659,4578,-11880},{1,42,1105,23310},20] (* Harvey P. Dale, Dec 14 2021 *)

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,9), n >= 3. - Milan Janjic, Apr 26 2009
a(n) = 42*a(n-1) - 659*a(n-2) + 4578*a(n-3) - 11880*a(n-4), n >= 4. - Vincenzo Librandi, Mar 18 2011
a(n) = 23*a(n-1) - 132*a(n-2) + 10^(n+1) - 9^(n+1), n >= 2. - Vincenzo Librandi, Mar 18 2011
a(n) = 5*10^(n+2) + 2*12^(n+2) - 11^(n+3)/2 - 3*9^(n+2)/2. - R. J. Mathar, Mar 19 2011

A016109 Expansion of 1/((1-7x)(1-8x)(1-9x)(1-10*x)).

Original entry on oeis.org

1, 34, 725, 12410, 186501, 2571114, 33339685, 413066170, 4941549701, 57504755594, 654463491045, 7314256515930, 80522026412101, 875355238834474, 9415203971344805, 100355146006589690

Offset: 0

Robert G. Wilson v

nonn

Index entries for linear recurrences with constant coefficients, signature (34,-431,2414,-5040).

Cf. A000453, A025211, A028025, A003468, A028165, A028200, A016075, A016094.

CoefficientList[Series[1/((1-7x)(1-8x)(1-9x)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[{34,-431,2414,-5040},{1,34,725,12410},21] (* Harvey P. Dale, Jan 26 2012 *)

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,7), n >= 3. - Milan Janjic, Apr 26 2009; adapted by R. J. Mathar, Mar 15 2011
a(n) = 19*a(n-1) - 90*a(n-2) + 8^(n+1) - 7^(n+1), n >= 2. - Vincenzo Librandi, Mar 12 2011
a(n) = (10^(n+3) - 3*9^(n+3) + 3*8^(n+3) - 7^(n+3))/6.  - Bruno Berselli, Mar 12 2011
a(n) = 34*a(n-1) - 431*a(n-2) + 2414*a(n-3) - 5040*a(n-4); a(0)=1, a(1)=34, a(2)=725, a(3)=12410. - Harvey P. Dale, Jan 26 2012

Offset changed to 0 by Vincenzo Librandi, Mar 12 2011

A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 19, 8, 1, 14, 55, 65, 16, 1, 20, 125, 285, 211, 32, 1, 27, 245, 910, 1351, 665, 64, 1, 35, 434, 2380, 5901, 6069, 2059, 128, 1, 44, 714, 5418, 20181, 35574, 26335, 6305, 256, 1, 54, 1110, 11130, 58107, 156660, 204205, 111645, 19171, 512

Offset: 1

Thomas Wieder, Dec 29 2006

Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row A000079 = the powers of two = 2^n. As the second row we have A001047 = 3^n - 2^n. As the column sums we have 1,3,10,37,151,674,3263,17007,94828 we have A005493 = number of partitions of [n+1] with a distinguished block.

			Matrix begins:
1, 2, 4,  8, 16,  32,   64,  128,   256, ... A000079
0, 1, 5, 19, 65, 211,  665, 2059,  6305, ... A001047
0, 0, 1,  9, 55, 285, 1351, 6069, 26335, ... A016269
0, 0, 0,  1, 14, 125,  910, 5901, 35574, ... A025211
0, 0, 0,  0,  1,  20,  245, 2380, 20181, ...
0, 0, 0,  0,  0,   1,   27,  434,  5418, ...
0, 0, 0,  0,  0,   0,    1,   35,   714, ...
0, 0, 0,  0,  0,   0,    0,    1,    44, ...
0, 0, 0,  0,  0,   0,    0,    0,     1, ...
Triangle begins:
1;
1,  2;
1,  5,  4;
1,  9, 19,  8;
1, 14, 55, 65, 16;

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A039810, A039814, A126350, A054654, A126353.

T:= (n, k)-> add(binomial(n-1, i-1) *Stirling2(i, n+1-k), i=1..n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 29 2011

T[n_, k_] := Sum[Binomial[n-1, i-1]*StirlingS2[i, n+1-k], {i, 1, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

(In Maple notation:) Matrix product B.A of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling2(j,i) with i from 1 to d, j from 1 to d, d=9.

T(n,k) = Sum_{i=1..n} C(n-1,i-1) * Stirling2(i, n+1-k). - Alois P. Heinz, Sep 29 2011

A099111 Expansion of 1 / ((1+x)(1-2x)(1-3x)(1-4x)(1-5x)).

Original entry on oeis.org

1, 13, 112, 798, 5103, 30471, 173734, 958936, 5170165, 27396369, 143295516, 742128114, 3814432987, 19490910907, 99140278258, 502476527532, 2539579950369, 12806979393285, 64472086878760, 324111808432390

Offset: 0

Ralf Stephan, Sep 28 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13, -57, 83, 34, -120).

First differences are in A004058. Pairwise sums are in A025211.

[(1/360)*(6250*5^n - 9216*4^n + 3645*3^n - 320*2^n + (-1)^n): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

CoefficientList[Series[1/((1+x)(1-2x)(1-3x)(1-4x)(1-5x)),{x,0,20}],x] (* or *) LinearRecurrence[{13,-57,83,34,-120},{1,13,112,798,5103},20] (* Harvey P. Dale, Jan 17 2024 *)

a(n) = (1/360)*(6250*5^n - 9216*4^n + 3645*3^n - 320*2^n + (-1)^n).

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 19, 9, 1, 0, 16, 65, 55, 14, 1, 0, 32, 211, 285, 125, 20, 1, 0, 64, 665, 1351, 910, 245, 27, 1, 0, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 0, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1

Offset: 0

Peter Luschny, Apr 10 2016

			1,
0, 1,
0, 2, 1,
0, 4, 5, 1,
0, 8, 19, 9, 1,
0, 16, 65, 55, 14, 1,
0, 32, 211, 285, 125, 20, 1,
0, 64, 665, 1351, 910, 245, 27, 1.

Peter Luschny, Extensions of the binomial

Variant: A143494 (the main entry for this triangle).
A005493 (row sums), A074051 (alt. row sums), A000079 (col. 1), A001047 (col. 2),
A016269 (col. 3), A025211 (col. 4), A000096 (diag. n,n-1), A215862 (diag. n,n-2),
A049444, A136124, A143491 (matrix inverse).
Cf. A048993, A269951.

A269952 := (n,k) -> Stirling2(n+1, k+1) - Stirling2(n, k+1):
seq(seq(A269952(n,k), k=0..n), n=0..9);

Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] StirlingS2[j,k], {j,0,n}], {n,0,9}, {k,0,n}]]

T(n, k) = S2(n+1, k+1) - S2(n, k+1).

cp's OEIS Frontend

A143494 Triangle read by rows: 2-Stirling numbers of the second kind.

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A028025 Expansion of 1/((1-3x)*(1-4x)*(1-5x)*(1-6x)).

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A016075 Expansion of 1/((1-8*x)*(1-9*x)*(1-10*x)*(1-11*x)).

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A028165 Expansion of 1/((1-5x)*(1-6x)*(1-7x)*(1-8x)).

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A028200 Expansion of 1/((1-6x)*(1-7x)*(1-8x)*(1-9x)).

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A016094 Expansion of 1/((1-9*x)*(1-10*x)*(1-11*x)*(1-12*x)).

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A016109 Expansion of 1/((1-7*x)*(1-8*x)*(1-9*x)*(1-10*x)).

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A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

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A028025 Expansion of 1/((1-3x)(1-4x)(1-5x)*(1-6x)).

A016075 Expansion of 1/((1-8x)(1-9x)(1-10x)(1-11*x)).

A028165 Expansion of 1/((1-5x)(1-6x)(1-7x)*(1-8x)).

A028200 Expansion of 1/((1-6x)(1-7x)(1-8x)*(1-9x)).

A016094 Expansion of 1/((1-9x)(1-10x)(1-11x)(1-12*x)).

A016109 Expansion of 1/((1-7x)(1-8x)(1-9x)(1-10*x)).

A099111 Expansion of 1 / ((1+x)(1-2x)(1-3x)(1-4x)(1-5x)).

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.