A000453 -id:A000453 - cp's OEIS frontend

A228839 Erroneous version of A000453.

0, 0, 0, 1, 20, 65, 350

Offset: 1

Olivier Gérard, Sep 05 2013 based on emails by Antreas P. Hatzipolakis (anopolis72 (AT) gmail.com) and R. J. Mathar

dead

This is column k=4 of the table on that page.

			The fifth term (20) is a typo for 10.
There is another error in this table in the book (entry for k=6, n=7 should be 21 instead of 20) leading to an incorrect row sum of 876 instead of 877 for the n=7 Bell number.

M. Barbut and B. Monjardet, Ordre et classification - algèbre et combinatoire - tome II, Série Méthodes Mathématiques des Sciences de l'Homme, Hachette, Paris, 1970, page 101.

Cf. A000453.

A016269 Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.

Original entry on oeis.org

1, 9, 55, 285, 1351, 6069, 26335, 111645, 465751, 1921029, 7859215, 31964205, 129442951, 522538389, 2104469695, 8460859965, 33972448951, 136276954149, 546269553775, 2188563950925, 8764714059751, 35090233104309, 140455067207455, 562102681589085, 2249257981411351

Offset: 0

N. J. A. Sloane

Half the number of 2 X (n+2) binary arrays with both a path of adjacent 1's and a path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002
As (0,0,1,9,55,...) this is the third binomial transform of cosh(x)-1. It is the binomial transform of A000392, when this has two leading zeros. Its e.g.f. is then exp(3x)cosh(x) - exp(3x) and a(n) = (4^n - 2*3^n + 2^n)/2. - Paul Barry, May 13 2003
Let P(A) be the power set of an n-element set A. Then a(n-2) is the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x. - Ross La Haye, Jan 10 2008
a(n) also gives the third column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - Wolfdieter Lang, Oct 08 2011
a(n) is also the number of even binomial coefficients in rows 0 through 2^(n+1)-1 of Pascal's triangle. - Aaron Meyerowitz, Oct 29 2013

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
K. S. Brown, Dedekind's problem
John Elias, Illustration of Initial Terms: Inverse of the Sierpinski Triangle
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
N. M. Rivière, Recursive formulas on free distributive lattices, J. Combinatorial Theory 5 1968 229--234. MR0231764 (38 #92). - _N. J. A. Sloane_, May 12 2012
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Equals (1/2) A038721(n+1). First differences of A000453. Partial sums of A027650. Pairwise sums of A099110. Odd part of A019333.

Cf. A000079, A000392, A001047, A006516, A032263, A143494, A193685.

[(2^n)*(2^n-1)/2-3^n+2^n: n in [2..30]]; // Vincenzo Librandi, Oct 06 2017

a:= n-> Stirling2(n+4, 4)-Stirling2(n+3, 4): seq(a(n), n=0..24); # Zerinvary Lajos, Oct 05 2007

CoefficientList[1/((1-2x)(1-3x)(1-4x)) + O[x]^30, x] (* Jean-François Alcover, Nov 28 2015 *)
LinearRecurrence[{9, -26, 24}, {1, 9, 55}, 40] (* Vincenzo Librandi, Oct 06 2017 *)

a(n)=(2^n)*(2^n-1)/2-3^n+2^n \\ Charles R Greathouse IV, Mar 22 2016

G.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)).
a(n-2) = (2^n)*(2^n - 1)/2 - 3^n + 2^n.
From Hieronymus Fischer, Jun 25 2007: (Start)
a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*4^k.
a(n) = 2^(n+1)*(1+2^(n+2))-3^(n+2). (End)
a(n) = 3*StirlingS2(n+3,4) + StirlingS2(n+3,3). - Ross La Haye, Jan 10 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,2,2), (n >= 2). - Milan Janjic, Apr 26 2009
E.g.f.: (d^2/dx^2) (exp(2*x)*((exp(x)-1)^2)/2!). See the Sheffer comment given above. - Wolfdieter Lang, Oct 08 2011
a(n) = A006516(n+2) - A001047(n+2). - Ross La Haye, Jan 26 2016
a(n) = A006516(n+1) + 3*a(n-1), n>=1, a(0)=1. - Carlos A. Rico A., Jun 22 2019

A032263 Number of ways to partition n labeled elements into 4 pie slices allowing the pie to be turned over; number of 2-element proper antichains of an n-element set.

Original entry on oeis.org

0, 0, 0, 3, 30, 195, 1050, 5103, 23310, 102315, 437250, 1834503, 7597590, 31175235, 127067850, 515396703, 2083011870, 8396420955, 33779000850, 135696347703, 544527210150, 2183335871475, 8749027724250, 35043169903503, 140313869216430, 561679070838795

Offset: 1

Christian G. Bower

A proper antichain is an antichain iff each two of its members have a nonempty intersection.

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which x and y are intersecting but for which x is not a subset of y and y is not a subset of x. This is just a different formulation of the alternative sequence description. - Ross La Haye, Jan 09 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. G. Bower, Transforms (2)
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A000453.

I:=[0,0,0,3]; [n le 4 select I[n] else 10*Self(n-1)-35*Self(n-2)+50*Self(n-3)-24*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Oct 19 2013

A032263 := proc(n) (4^n-4*3^n+6*2^n-4)/8 ; end: seq(A032263(n),n=1..20) ; # R. J. Mathar, Feb 26 2008

CoefficientList[Series[(3x^4)/((1-x)(1-2x)(1-3x)(1-4x)),{x,0,40}],x] (* Harvey P. Dale, Feb 28 2013 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -24,50,-35,10]^(n-1)*[0;0;0;3])[1,1] \\ Charles R Greathouse IV, Feb 09 2017

"DIJ[ 4 ]" (bracelet, indistinct, labeled, 4 parts) transform of 1, 1, 1, 1, ...
3*S(n,4) = (4^n-4*3^n+6*2^n-4)/8. - R. J. Mathar, Feb 26 2008
G.f.: 3*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). - Colin Barker, May 29 2012
a(n) = 3*A000453(n). - Alois P. Heinz, Jan 24 2018
E.g.f.: (exp(x) - 1)^4/8. - Stefano Spezia, Apr 06 2022

Alternative description from Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

More terms from Vincenzo Librandi, Oct 19 2013

A028244 a(n) = 4^(n-1) - 33^(n-1) + 32^(n-1) - 1 (essentially Stirling numbers of second kind).

Original entry on oeis.org

0, 0, 0, 6, 60, 390, 2100, 10206, 46620, 204630, 874500, 3669006, 15195180, 62350470, 254135700, 1030793406, 4166023740, 16792841910, 67558001700, 271392695406, 1089054420300, 4366671742950, 17498055448500, 70086339807006

Offset: 1

N. J. A. Sloane, Doug McKenzie (mckfam4(AT)aol.com)

For n>=4, a(n) is equal to the number of functions f: {1,2,...,n-1}->{1,2,3,4} such that Im(f) contains 3 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 27 2007

Seiichi Manyama, Table of n, a(n) for n = 1..1661
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, Electronics Letters ( Volume: 50, Issue: 1, January 2 2014 ).
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (10, -35, 50, -24).

Cf. A000453, A008277, A032263, A163626.

[4^(n-1) - 3*3^(n-1) + 3*2^(n-1) - 1: n in [1..30]]; // G. C. Greubel, Nov 19 2017

Table[4^(n - 1) - 3*3^(n - 1) + 3*2^(n - 1) - 1, {n, 1, 30}] (* Stefan Steinerberger, Apr 13 2006 *)
Table[6*StirlingS2[n,4], {n,1,30}] (* G. C. Greubel, Nov 19 2017 *)

for(n=1,30, print1(6*stirling(n,4,2), ", ")) \\ G. C. Greubel, Nov 19 2017

a(n) = 6*S(n, 4) = 6*A000453(n). - Emeric Deutsch, May 02 2004
G.f.: 6x^4/((1-x)(1-2x)(1-3x)(1-4x)). - R. J. Mathar, Oct 23 2008
E.g.f.: (exp(4*x) - 4*exp(3*x) + 6*exp(2*x) - 4*exp(x) + 1)/4, with a(0) = 0. - Wolfdieter Lang, May 03 2017
a(n) = 2*A032263(n). - Alois P. Heinz, Jan 24 2018

A000919 a(n) = 4^n - C(4,3)3^n + C(4,2)2^n - C(4,1).

Original entry on oeis.org

0, 0, 0, 24, 240, 1560, 8400, 40824, 186480, 818520, 3498000, 14676024, 60780720, 249401880, 1016542800, 4123173624, 16664094960, 67171367640, 270232006800, 1085570781624, 4356217681200, 17466686971800, 69992221794000, 280345359228024, 1122510953731440

Offset: 1

N. J. A. Sloane

Differences of 0: 4!*S(n,4).
Number of surjections from an n-element set onto a four-element set. - David Wasserman, Jun 06 2007
Number of rows of n colors using exactly four colors.  For n=4, the 24 rows are the 24 permutations of ABCD. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, http://www.exp-math.uni-essen.de/~immink/pdf/jsac13.pdf, 2013. [This link no longer works, but please do not delete this reference, for historical reasons. Michel Marcus has suggested that the Immink link below points to the published version of the original reference, and I agree. - N. J. A. Sloane, May 29 2023]
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.

T. D. Noe, Table of n, a(n) for n = 1..201
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, Electronics Letters 50(1):20-22, January 2014.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Ethan Soloway, Megan Triplett, and Wenshi Zhao, On the Existence of Partition of the Hypercube Graph into 3 Initial Segments, arXiv:2501.05827 [math.CO], 2025. See p. 23.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A001117, A001118.

Column 4 of A019538.

with (combstruct):ZL:=[S,{S=Sequence(U,card=r),U=Set(Z,card>=1)}, labeled]: seq(count(subs(r=4,ZL),size=m),m=1..25); # Zerinvary Lajos, Mar 09 2007
A000919:=24/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation

nn = 25; CoefficientList[Series[24 x^3/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x)), {x, 0, nn}], x] (* T. D. Noe, Jun 20 2012 *)
k=4; Table[k!StirlingS2[n,k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 4!*stirling(n, 4, 2); \\ Altug Alkan, Sep 25 2018

G.f.: 24*x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).
a(n) = 4^n - binomial(4,3)*3^n + binomial(4,2)*2^n - binomial(4,1) = 24*A000453(n). - David Wasserman, Jun 06 2007
E.g.f.: (exp(x)-1)^4. - Geoffrey Critzer, Feb 11 2009
For n >= 4: a(n+1) = 4*a(n) + 4*(3^n - 3*2^n + 3) = 4*a(n) + 4*A001117(n). - Geoffrey Critzer, Feb 27 2009
a(n) = k!*S2(n,k), where k=4 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A346895 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^4 / 4!).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 65, 350, 1771, 10290, 86605, 977350, 11778041, 138208070, 1590920695, 18895490250, 245692484311, 3587464083850, 57397496312585, 966066470023550, 16713560617838581, 297182550111615630, 5500448659383161275, 107267326981597659250

Offset: 0

Ilya Gutkovskiy, Aug 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..461

Cf. A000670, A330047, A346894, A346920.

Cf. A000453, A327505, A346923, A353775.

nmax = 23; CoefficientList[Series[1/(1 - (Exp[x] - 1)^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^4/4!))) \\ Michel Marcus, Aug 06 2021

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/(24^k*prod(j=1, 4*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 4, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022

a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/24^k); \\ Seiichi Manyama, May 07 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).
a(n) ~ n! / (4*(1 + 2^(-3/4)*3^(-1/4)) * log(1 + 2^(3/4)*3^(1/4))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/(24^k * Product_{j=1..4*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/24^k. (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

cp's OEIS Frontend

A228839 Erroneous version of A000453.

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A016269 Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.

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A032263 Number of ways to partition n labeled elements into 4 pie slices allowing the pie to be turned over; number of 2-element proper antichains of an n-element set.

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A028244 a(n) = 4^(n-1) - 3*3^(n-1) + 3*2^(n-1) - 1 (essentially Stirling numbers of second kind).

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A000919 a(n) = 4^n - C(4,3)*3^n + C(4,2)*2^n - C(4,1).

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A346895 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^4 / 4!).

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

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A028244 a(n) = 4^(n-1) - 33^(n-1) + 32^(n-1) - 1 (essentially Stirling numbers of second kind).

A000919 a(n) = 4^n - C(4,3)3^n + C(4,2)2^n - C(4,1).

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)).