A028025 -id:A028025 - cp's OEIS frontend

A143495 Triangle read by rows: 3-Stirling numbers of the second kind.

1, 3, 1, 9, 7, 1, 27, 37, 12, 1, 81, 175, 97, 18, 1, 243, 781, 660, 205, 25, 1, 729, 3367, 4081, 1890, 380, 33, 1, 2187, 14197, 23772, 15421, 4550, 644, 42, 1, 6561, 58975, 133057, 116298, 47271, 9702, 1022, 52, 1, 19683, 242461, 724260, 830845, 447195

Offset: 3

Peter Bala, Aug 20 2008

This is the case r = 3 of the r-Stirling numbers of the second kind. The 3-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, 2 and 3 belong to distinct subsets. For remarks on the general case see A143494 (r = 2). The corresponding array of 3-Stirling numbers of the first kind is A143492. The theory of r-Stirling numbers of both kinds is developed in [Broder]. For 3-Lah numbers refer to A143498.
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^(-3)*E^n*x^3 = Sum_{k = 0..n} T(n+3,k+3)*x^k*D^k.
The row generating polynomials R_n(x) := Sum_{k= 3..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_3(x) = x^3. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).
Relation with the 3-Eulerian numbers E_3(n,j) := A144697(n,j): T(n,k) = (3!/k!)*Sum_{j = n-k..n-3} E_3(n,j)*binomial(j,n-k) for n >= k >= 3.
(End)
T(n,k) = S(n,k,3), n>=k>=3, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column k from (A20) with k->3, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(3*x),exp(x)-1) with e.g.f. of column no. m>=0: exp(3*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. - Wolfdieter Lang, Sep 29 2011

			Triangle begins
  n\k|....3....4....5....6....7....8
  ==================================
  3..|....1
  4..|....3....1
  5..|....9....7....1
  6..|...27...37...12....1
  7..|...81..175...97...18....1
  8..|..243..781..660..205...25....1
  ...
T(5,4) = 7. The set {1,2,3,4,5} can be partitioned into four subsets such that 1, 2 and 3 belong to different subsets in 7 ways: {{1}{2}{3}{4,5}}, {{1}{2}{5}{3,4}}, {{1}{2}{4}{3,5}}, {{1}{3}{4}{2,5}}, {{1}{3}{5}{2,4}}, {{2}{3}{4}{1,5}} and {{2}{3}{5}{1,4}}.
From _Peter Bala_, Feb 23 2025: (Start)
The array factorizes as
/ 1               \       /1             \ /1             \ /1            \
| 3    1           |     | 3   1          ||0  1           ||0  1          |
| 9    7   1       |  =  | 9   4   1      ||0  3   1       ||0  0  1       | ...
|27   37  12   1   |     |27  13   5  1   ||0  9   4  1    ||0  0  3  1    |
|81  175  97  18  1|     |81  40  18  6  1||0 27  13  5  1 ||0  0  9  4  1 |
|...               |     |...             ||...            ||...           |
where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 3*x), x/(1 - x)). See A106516. (End)

Peter Bala, Factorising (r,b)-Stirling arrays
Andrei Z. Broder, The r-Stirling numbers, Report Number: CS-TR-82-949, Stanford University, Department of Computer Science; see also, Discrete Math. 49, 241-259 (1984).
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [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.
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, Discrete Math. 239 No. 1-3, 33-51 (2001).
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.

Cf. A005061 (column 4), A005494 (row sums), A008277, A016753 (column 5), A028025 (column 6), A049458 (matrix inverse), A106516, A143492, A143494, A143496, A143498.

A143495 := (n, k) -> (1/(k-3)!)*add((-1)^(k-i-1)*binomial(k-3,i)*(i+3)^(n-3), i = 0..k-3): for n from 3 to 12 do seq(A143495(n, k), k = 3..n) end do;

nmax = 12; t[n_, k_] := 1/(k-3)!* Sum[ (-1)^(k-j-1)*Binomial[k-3, j]*(j+3)^(n-3), {j, 0, k-3}]; Flatten[ Table[ t[n, k], {n, 3, nmax}, {k, 3, n}]] (* Jean-François Alcover, Dec 07 2011, after Maple *)

@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)
A143495 = lambda n, k: stirling2r(n, k, 3)
for n in (3..8): [A143495(n, k) for k in (3..n)] # Peter Luschny, Nov 19 2012

T(n+3,k+3) = (1/k!)*Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+3)^n, n,k >= 0.
T(n,k) = Stirling2(n,k) - 3*Stirling2(n-1,k) + 2*Stirling2(n-2,k), n,k >= 3.
Recurrence relation: T(n,k) = T(n-1,k-1) + k*T(n-1,k) for n > 3, with boundary conditions: T(n,2) = T(2,n) = 0 for all n; T(3,3) = 1; T(3,k) = 0 for k > 3.
Special cases: T(n,3) = 3^(n-3); T(n,4) = 4^(n-3) - 3^(n-3).
E.g.f. (k+3) column (with offset 3): (1/k!)*exp(3x)*(exp(x)-1)^k.
O.g.f. k-th column: Sum_{n >= k} T(n,k)*x^n = x^k/((1-3*x)*(1-4*x)*...*(1-k*x)).
E.g.f.: exp(3*t + x*(exp(t)-1)) = Sum_{n >= 0} Sum_{k = 0..n} T(n+3,k+3)*x^k*t^n/n! = Sum_{n >= 0} B_n(3;x)*t^n/n! = 1 + (3+x)*t/1! + (9+7*x+x^2)*t^2/2! + ..., where the row polynomials, B_n(3;x) := Sum_{k = 0..n} T(n+3,k+3)*x^k, may be called the 3-Bell polynomials.
Dobinski-type identities: Row polynomial B_n(3;x) = exp(-x)*Sum_{i >= 0} (i+3)^n*x^i/i!; Sum_{k = 0..n} k!*T(n+3,k+3)*x^k = Sum_{i >= 0} (i+3)^n*x^i/(1+x)^(i+1).
The T(n,k) are the connection coefficients between the falling factorials and the shifted monomials (x+3)^(n-3). For example, 9 + 7*x + x*(x-1) = (x+3)^2 and 27 + 37*x + 12x*(x-1) + x*(x-1)*(x-2) = (x+3)^3.
This array is the matrix product P^2 * S, where P denotes Pascal's triangle, A007318 and S denotes the lower triangular array of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]). The inverse array is A049458, the signed 3-Stirling numbers of the first kind.

A143395 Triangle read by rows: T(n,k) = number of forests of k labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 7, 9, 1, 0, 15, 55, 18, 1, 0, 31, 285, 205, 30, 1, 0, 63, 1351, 1890, 545, 45, 1, 0, 127, 6069, 15421, 7770, 1190, 63, 1, 0, 255, 26335, 116298, 95781, 24150, 2282, 84, 1, 0, 511, 111645, 830845, 1071630, 416451, 62370, 3990, 108, 1

Offset: 0

Alois P. Heinz, Aug 12 2008

This is the Sheffer triangle (1,exp(x)*(exp(x)-1)) (Jobotinsky type). See the e.g.f. given by V. Jovovic below, and the W. Lang link under A006232 (second part) for general Sheffer remarks and the conversion to the umbral notation of S. Roman's book. - Wolfdieter Lang, Oct 08 2011
From Peter Bala, Jan 07 2015: (Start)
T(n,k) counts the ways a set of size n can be partitioned into k nonempty blocks and then a nonempty subset chosen from each block. An example is given below.
This triangle is the particular case a = 1, b = 1, c = 0 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. A008277 is the case a = 1, b = 0, c = 0.
Define a polynomial sequence x_(n) by putting x_(0) = 1, x_(1) = x and for n >= 2 setting x_(n) = x*(x - (n+1))*(x - (n+2))*...*(x - (2*n-1)), that is, x_(n) = (-1)^(n+1)*n!*(x/(2*n - x))*binomial(2*n - x,n) for n >= 0. Then this table is the triangle of connection constants for expressing the monomial polynomials x^n in terms of the basis polynomials x_(k), that is, x^n = sum {k = 0..n} T(n,k)*x_(k), n = 0,1,2,.... Examples are given below.
Matrix factorization: Let M be the infinite lower unit triangular array with (n,k)-th entry (2^(n+1-k)-1)*binomial(n,k). 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. It follows from the recurrence equation given in the Formula section that the infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle (but with the first row and column omitted). See the Example section. (End)
The Bell transform of 2^(n+1)-1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			T(3,2) = 9: {1}{2}<-3, {1}{3}<-2, {1}{2,3}, {2}{1}<-3, {2}{3}<-1, {2}{1,3}, {3}{1}<-2, {3}{2}<-1, {3}{1,2}.
Triangle begins:
  1;
  0,   1;
  0,   3,    1;
  0,   7,    9,     1;
  0,  15,   55,    18,    1;
  0,  31,  285,   205,   30,    1;
  0,  63, 1351,  1890,  545,   45,  1;
  0, 127, 6069, 15421, 7770, 1190, 63,  1;
  ...
From _Peter Bala_, Jan 07 2015: (Start)
T(4,2) = 55: There are 7 partitions of the set {1,2,3,4} into 2 blocks. For the 3 set partitions of the type {a,b}{c,d} we can choose a nonempty subset from each block in one of 3*3 ways giving 3*3*3 = 27 possibilities in all. The remaining 4 set partitions of {1,2,3,4} into 2 blocks are of the form {a,b,c}{d} and we can choose a nonempty subset from each block in 7*1 ways giving 4*7*1 = 28 possible choices. Thus in total T(4,2) = 27 + 28 = 55.
Recurrence equation example:
T(4,2) = sum {j = 1..3} (2^(4-j) - 1)*binomial(3,j)*T(j,1) = 7*3*1 + 3*3*3 + 1*1*7 = 55.
Connection constants:
Row 3 = [0, 7, 9, 1]. Hence x^3 = 7*x + 9*x*(x - 3) + x*(x - 4)*(x - 5); Row 4 = [0, 15, 55, 18, 1]. Hence x^4 = 15*x + 55*x*(x - 3) + 18*x*(x - 4)*(x - 5) + x*(x - 5)*(x - 6)*(x - 7).
With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/ 1        \/1           \/1       \       / 1        \
| 3  1     ||0  1        ||0 1      |      | 3  1      |
| 7  6  1  ||0  3  1     ||0 0 1    |... = | 7  9  1   |
|15 21 9 1 ||0  7  6  1  ||0 0 3 1  |      |15 55 18 1 |
|...       ||0 15 21  9 1||0 0 7 6 1|      |...        |
|...       ||...         ||...      |      |           |
(End)

Alois P. Heinz, Rows n = 0..140, flattened
Eli Bagno, Riccardo Biagioli, and David Garber, Some identities involving second kind Stirling numbers of types B and D, arXiv:1901.07830 [math.CO], 2019.
Peter Bala, A 3 parameter family of generalized Stirling numbers
Takao Komatsu, Eli Bagno, and David Garber, A q,r-analogue of poly-Stirling numbers of second kind with combinatorial applications, arXiv:2209.06674 [math.CO], 2022.
Index entries for sequences related to rooted trees

Columns k=0-9: A000007, A000225, A016269(n-2), A028025(n-3), A143399, A143400, A143401, A143402, A143403, A143404.
Diagonal: A000012.
T(2*n,n) gives A383869.
See also A048993, A008277, A007318, A143405 for row sums.

[[(&+[Binomial(n,j)*StirlingSecond(j,k)*k^(n-j): j in [k..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 07 2019

T:= (n, k)-> add(binomial(n,t)*Stirling2(t,k)*k^(n-t), t=k..n):
seq(seq(T(n, k), k=0..n), n=0..11);

t[0, 0]=1; t[n_, k_]:= SeriesCoefficient[Exp[y*Exp[x]*(Exp[x]-1)], {x, 0, n}, {y, 0, k}]*n!; Table[t[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 05 2013, after Vladeta Jovovic *)
Table[If[n==k==0, 1, If[k==0, 0, Sum[Binomial[n, j]*StirlingS2[j, k]* k^(n-j), {j,k,n}]]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 07 2019 *)

{T(n,k) = sum(j=k, n, binomial(n,j)*stirling(j,k,2)*k^(n-j))};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Mar 07 2019

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 2^(n+1)-1, 10) # Peter Luschny, Jan 18 2016

G.f. for column k: x^k/Product_{t=k..2*k} (1-t*x).
T(n,k) = Sum_{t=k..n} C(n,t) * Stirling2(t,k) * k^(n-t).
E.g.f.: exp(y*exp(x)*(exp(x)-1)). - Vladeta Jovovic, Dec 08 2008
T(n,k) = Sum_{m=0..k} Stirling2(n,k+m)*(k+m)!/(m!*(k-m)!). - Vladimir Kruchinin, Apr 06 2011
Let P be Pascal's triangle A007318. The first column of the array exp(t*(P^2-P)) gives the row generating polynomials of this triangle.
The row polynomials R(n,t) satisfy the recurrence R(n+1,t) = t*(Sum_{k = 0..n} (2^(k+1)-1)*C(n,k)*R(n-k,t)) with R(0,t) = 1. For example, the row 4 polynomial R(4,t) = 15*t + 55*t^2 + 18*t^3 + t^4 = t*((7*t + 9*t^2 + t^3) + 3*3*(3*t+t^2) + 7*3*t + 15*1). - Peter Bala, Oct 12 2011
From Peter Bala, Jan 07 2015: (Start)
T(n,k) = (1/k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(j + k)^n.
Recurrence equation: T(n+1,k+1) = Sum_{j = k..n} (2^(n-j+1) - 1)*binomial(n,j)*T(j,k) with T(0,0) = 1 and T(n,0) = 0 for n >= 1. This leads to the matrix factorization noted in the Comments section.
The inverse array is a signed version of A038455. (End)

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

cp's OEIS Frontend

A143495 Triangle read by rows: 3-Stirling numbers of the second kind.

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A143395 Triangle read by rows: T(n,k) = number of forests of k labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, n >= 0, 0 <= k <= n.

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

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Magma

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

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PARI

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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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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)).