A003468 -id:A003468 - cp's OEIS frontend

A028109 Duplicate of A003468.

1, 22, 305, 3410, 33621, 305382, 2619625, 21554170, 171870941

Offset: 0

dead

A143496 4-Stirling numbers of the second kind.

1, 4, 1, 16, 9, 1, 64, 61, 15, 1, 256, 369, 151, 22, 1, 1024, 2101, 1275, 305, 30, 1, 4096, 11529, 9751, 3410, 545, 39, 1, 16384, 61741, 70035, 33621, 7770, 896, 49, 1, 65536, 325089, 481951, 305382, 95781, 15834, 1386, 60, 1, 262144, 1690981, 3216795

Offset: 4

Peter Bala, Aug 20 2008

This is the case r = 4 of the r-Stirling numbers of the second kind. The 4-Stirling numbers of the second kind count the ways of partitioning the set {1,2,...,n} into k nonempty disjoint subsets with the restriction that the elements 1, 2, 3 and 4 belong to distinct subsets. For remarks on the general case see A143494 (r = 2). The corresponding array of 4-Stirling numbers of the first kind is A143493. The theory of r-Stirling numbers of both kinds is developed in [Broder]. For 4-Lah numbers refer to A143499.
From Wolfdieter Lang, Sep 29 2011: (Start)
T(n,k) = S(n,k,4), n >= k >= 4, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column k from (A20) with k->4, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(4*x),exp(x)-1) with e.g.f. of column no. m >= 0: exp(4*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|.....4.....5.....6.....7.....8.....9
========================================
4..|.....1
5..|.....4.....1
6..|....16.....9.....1
7..|....64....61....15.....1
8..|...256...369...151....22.....1
9..|..1024..2101..1275...305....30.....1
...
T(6,5) = 9. The set {1,2,3,4,5,6} can be partitioned into five subsets such that 1, 2, 3 and 4 belong to different subsets in 9 ways: {{1,5}{2}{3}{4}{6}}, {{1,6}{2}{3}{4}{5}}, {{2,5}{1}{3}{4}{6}}, {{2,6}{1}{3}{4}{5}}, {{3,5}{1}{2}{4}{6}}, {{3,6}{1}{2}{4}{5}}, {{4,5}{1}{2}{3}{6}}, {{4,6}{1}{2}{3}{5}} and {{5,6}{1}{2}{3}{4}}.

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).
Askar Dzhumadil'daev and Damir 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.
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.
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)
Lily L. Liu and Yi 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. A003468 (column 7), A005060 (column 5), A008277, A016103 (column 6), A045379 (row sums), A049459 (matrix inverse), A143493, A143494, A143495, A143499.

with combinat: T := (n, k) -> 1/(k-4)!*add ((-1)^(k-i)*binomial(k-4,i)*(i+4)^(n-4),i = 0..k-4): for n from 4 to 13 do seq(T(n, k), k = 4..n) end do;

t[n_, k_] := StirlingS2[n, k] - 6*StirlingS2[n-1, k] + 11*StirlingS2[n-2, k] - 6*StirlingS2[n-3, k]; Flatten[ Table[ t[n, k], {n, 4, 13}, {k, 4, n}]] (* Jean-François Alcover, Dec 02 2011 *)

T(n+4,k+4) = (1/k!)*Sum_{i = 0..k} (-1)^(k-i)*C(k,i)*(i+4)^n, n,k >= 0.
T(n,k) = Stirling2(n,k) - 6*Stirling2(n-1,k) + 11*Stirling2(n-2,k) - 6*Stirling2(n-3,k) for n,k >= 4.
Recurrence relation: T(n,k) = T(n-1,k-1) + k*T(n-1,k) for n > 4 with boundary conditions: T(n,3) = T(3,n) = 0 for all n; T(4,4) = 1; T(4,k) = 0 for k > 4. Special cases: T(n,4) = 4^(n-4); T(n,5) = 5^(n-4) - 4^(n-4).
E.g.f. (k+4)-th column (with offset 4): (1/k!)*exp(4*x)*(exp(x)-1)^k.
O.g.f. k-th column: Sum_{n>=k} T(n,k)*x^n = x^k/((1-4*x)*(1-5*x)*...*(1-k*x)).
E.g.f.: exp(4*t + x*(exp(t)-1)) = Sum_{n = 0..infinity} Sum_(k = 0..n) T(n+4,k+4)*x^k*t^n/n! = Sum_{n = 0..infinity} B_n(4;x)*t^n/n! = 1 + (4+x)*t/1! + (16+9*x+x^2)*t^2/2! + ..., where the row polynomials, B_n(4;x) := Sum_{k = 0..n} T(n+4,k+4)*x^k, may be called the 4-Bell polynomials.
Dobinski-type identities: Row polynomial B_n(4;x) = exp(-x)*Sum_{i = 0..infinity} (i+4)^n*x^i/i!; Sum_{k = 0..n} k!*T(n+4,k+4)*x^k = Sum_{i = 0..infinity} (i+4)^n*x^i/(1+x)^(i+1).
The T(n,k) are the connection coefficients between the falling factorials and the shifted monomials (x+4)^(n-4). For example, 16 + 9*x + x*(x-1) = (x+4)^2; 64 + 61*x + 15*x*(x-1) + x*(x-1)*(x-2) = (x+4)^3.
This array is the matrix product P^3 * 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 A049459, the signed 4-Stirling numbers of the first kind.
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^(-4)*E^n*x^4 = Sum_{k = 0..n} T(n+4,k+4)*x^k*D^k.
The row generating polynomials R_n(x) := Sum_{k=4..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_4(x) = x^4. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).
Relation with the 4-Eulerian numbers E_4(n,j) := A144698(n,j): T(n,k) = 4!/k!*Sum_{j = n-k..n-4} E_4(n,j)*binomial(j,n-k) for n >= k >= 4.
(End)

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

A016103 Expansion of 1/((1-4x)(1-5x)(1-6x)).

Original entry on oeis.org

1, 15, 151, 1275, 9751, 70035, 481951, 3216795, 20991751, 134667555, 852639151, 5343198315, 33212784151, 205111785075, 1260114546751, 7708980203835, 46999640806951, 285743822630595, 1733261544204751

Offset: 0

Robert G. Wilson v

2*a(n-2) = 6^n - 2*5^n + 4^n is the number of 3 X n {0,1}-matrices such that: (a) first and second row have a common 1, (b) first and third row have a common 1, (c) second and third row have no common 1. - Andi Fugard and Vladeta Jovovic, Jul 26 2008

This is the third column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See A193685 for general comments. - Wolfdieter Lang, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Andi Fugard, Counting first-order models (with n individuals) of syllogisms.
Index entries for linear recurrences with constant coefficients, signature (15,-74,120).

Cf. A051588, A000302, A005060, A003468.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-5*x)*(1-6*x)))); // Vincenzo Librandi, Jun 24 2013

I:=[1, 15, 151]; [n le 3 select I[n] else 15*Self(n-1)-74*Self(n-2)+120*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 24 2013

CoefficientList[Series[1 / ((1 - 4 x) (1 - 5 x) (1 - 6 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 24 2013 *)

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

a(n) = 2^(3 + 2*n) + 2^(1 + n) * 3^(2 + n) - 5^(2 + n). - Andi Fugard, Jul 22 2008
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,4), n >= 2. - Milan Janjic, Apr 26 2009
O.g.f.: 1/((1-4*x)*(1-5*x)*(1-6*x)).
E.g.f.: (d^2/dx^2)(exp(4*x)*((exp(x)-1)^2)/2!). See the Sheffer triangle comment above. - Wolfdieter Lang, Oct 08 2011
a(n) = 15*a(n-1) - 74*a(n-2) + 120*a(n-3). - Vincenzo Librandi, Jun 24 2013

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

A057964 Triangle T(n,k) of number of minimal 3-covers of a labeled n-set that cover k points of that set uniquely (k=3,..,n).

Original entry on oeis.org

1, 16, 6, 160, 120, 25, 1280, 1440, 600, 90, 8960, 13440, 8400, 2520, 301, 57344, 107520, 89600, 40320, 9632, 966, 344064, 774144, 806400, 483840, 173376, 34776, 3025, 1966080, 5160960, 6451200, 4838400, 2311680, 695520, 121000, 9330

Offset: 3

Vladeta Jovovic, Oct 17 2000

Row sums give A003468.

			[1], [16, 6], [160, 120, 25], [1280, 1440, 600, 90], ...; There are 305=160+120+25 minimal 3-covers of a labeled 5-set.

Eric Weisstein's World of Mathematics, Minimal cover

Cf. A035347, A057669 (unlabeled case), A057963, A057965-A057968.

Number of minimal m-covers of a labeled n-set that cover k points of that set uniquely is C(n, k)*S(k, m)*(2^m-m-1)^(n-k), where S(k, m) are Stirling numbers of the second kind.

cp's OEIS Frontend

A028109 Duplicate of A003468.

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A143496 4-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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A016103 Expansion of 1/((1-4x)(1-5x)(1-6x)).

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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)).