A028723 -id:A028723 - cp's OEIS frontend

A006918 a(n) = binomial(n+3, 3)/4 for odd n, n(n+2)(n+4)/24 for even n.

0, 1, 2, 5, 8, 14, 20, 30, 40, 55, 70, 91, 112, 140, 168, 204, 240, 285, 330, 385, 440, 506, 572, 650, 728, 819, 910, 1015, 1120, 1240, 1360, 1496, 1632, 1785, 1938, 2109, 2280, 2470, 2660, 2870, 3080, 3311, 3542, 3795, 4048, 4324, 4600, 4900, 5200, 5525, 5850, 6201, 6552, 6930

Offset: 0

N. J. A. Sloane

Maximal number of inconsistent triples in a tournament on n+2 nodes [Kac]. - corrected by Leen Droogendijk, Nov 10 2014
a(n-4) is the number of aperiodic necklaces (Lyndon words) with 4 black beads and n-4 white beads.
a(n-3) is the maximum number of squares that can be formed from n lines, for n>=3. - Erich Friedman; corrected by Leen Droogendijk, Nov 10 2014
Number of trees with diameter 4 where at most 2 vertices 1 away from the graph center have degree > 2. - Jon Perry, Jul 11 2003
a(n+1) is the number of partitions of n into parts of two kinds, with at most two parts of each kind. Also a(n-3) is the number of partitions of n with Durfee square of size 2. - Franklin T. Adams-Watters, Jan 27 2006
Factoring the g.f. as x/(1-x)^2 times 1/(1-x^2)^2 we find that the sequence equals (1, 2, 3, 4, ...) convolved with (1, 0, 2, 0, 3, 0, 4, ...), A000027 convolved with its aerated variant. -  Gary W. Adamson, May 01 2009
Starting with "1" = triangle A171238 * [1,2,3,...]. - Gary W. Adamson, Dec 05 2009
The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of this sequence, e.g., Kn22(n) = a(n+1) + a(n) + 2*a(n-1) + a(n-2) and Fi2(n) = a(n) + 4*a(n-1) + a(n-2). - Johannes W. Meijer, May 20 2011
For n>3, a(n-4) is the number of (w,x,y,z) having all terms in {1,...,n} and w+x+y+z=|x-y|+|y-z|. - Clark Kimberling, May 23 2012
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w+x+y < |w-x|+|x-y|. - Clark Kimberling, Jun 13 2012
For n>0 number of inequivalent (n-1) X 2 binary matrices, where equivalence means permutations of rows or columns or the symbol set. - Alois P. Heinz, Aug 17 2014
Number of partitions p of n+5 such that p[3] = 2. Examples: a(1)=1 because we have (2,2,2); a(2)=2 because we have (2,2,2,1) and (3,2,2); a(3)=5 because we have (2,2,2,1,1), (2,2,2,2), (3,2,2,1), (3,3,2), and (4,2,2). See the R. P. Stanley reference. - Emeric Deutsch, Oct 28 2014
Sum over each antidiagonal of A243866. - Christopher Hunt Gribble, Apr 02 2015
Number of nonisomorphic outer planar graphs of order n>=3, size n+2, and maximum degree 3. - Christian Barrientos and Sarah Minion, Feb 27 2018
a(n) is the number of 2413-avoiding odd Grassmannian permutations of size n+1. - Juan B. Gil, Mar 09 2023

			G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 14*x^5 + 20*x^6 + 30*x^7 + 40*x^8 + 55*x^9 + ...
From _Gus Wiseman_, Apr 06 2019: (Start)
The a(4 - 3) = 1 through a(8 - 3) = 14 integer partitions with Durfee square of length 2 are the following (see Franklin T. Adams-Watters's second comment). The Heinz numbers of these partitions are given by A325164.
  (22)  (32)   (33)    (43)     (44)
        (221)  (42)    (52)     (53)
               (222)   (322)    (62)
               (321)   (331)    (332)
               (2211)  (421)    (422)
                       (2221)   (431)
                       (3211)   (521)
                       (22111)  (2222)
                                (3221)
                                (3311)
                                (4211)
                                (22211)
                                (32111)
                                (221111)
The a(0 + 1) = 1 through a(4 + 1) = 14 integer partitions of n into parts of two kinds with at most two parts of each kind are the following (see Franklin T. Adams-Watters's first comment).
  ()()  ()(1)  ()(2)   ()(3)    ()(4)
        (1)()  (2)()   (3)()    (4)()
               ()(11)  (1)(2)   (1)(3)
               (1)(1)  ()(21)   ()(22)
               (11)()  (2)(1)   (2)(2)
                       (21)()   (22)()
                       (1)(11)  ()(31)
                       (11)(1)  (3)(1)
                                (31)()
                                (11)(2)
                                (1)(21)
                                (2)(11)
                                (21)(1)
                                (11)(11)
The a(6 - 5) = 1 through a(10 - 5) = 14 integer partitions whose third part is 2 are the following (see Emeric Deutsch's comment). The Heinz numbers of these partitions are given by A307373.
  (222)  (322)   (332)    (432)     (442)
         (2221)  (422)    (522)     (532)
                 (2222)   (3222)    (622)
                 (3221)   (3321)    (3322)
                 (22211)  (4221)    (4222)
                          (22221)   (4321)
                          (32211)   (5221)
                          (222111)  (22222)
                                    (32221)
                                    (33211)
                                    (42211)
                                    (222211)
                                    (322111)
                                    (2221111)
(End)

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.
M. Kac, An example of "counting without counting", Philips Res. Reports, 30 (1975), 20*-22* [Special issue in honour of C. J. Bouwkamp].
E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, 2004.
K. B. Reid and L. W. Beineke "Tournaments", pp. 169-204 in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, p. 186, Theorem 6.11.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 2nd ed., 2012, Exercise 4.16, pp. 530, 552.
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.

T. D. Noe, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020.
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Yuriy Choliy and Andrew V. Sills, A formula for the partition function that “counts”, Preprint 2015.
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
S. J. Cyvin et al., Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
B. G. Eke, Monotonic triads, Discrete Math., Vol. 9, No. 4 (1974), pp. 359-363. MR0354390 (50 #6869)
Irene Erazo, John López, and Carlos Trujillo, A combinatorial problem that arose in integer B_3 Sets, Revista Colombiana de Matemáticas, Vol. 53, No. 2 (2019), pp. 195-203.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
John Golden and Marcus Spradlin, Collinear and Soft Limits of Multi-Loop Integrands in N= 4 Yang-Mills, arXiv preprint arXiv:1203.1915 [hep-th], 2012. - From _N. J. A. Sloane_, Sep 14 2012
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 11.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=2]
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).
Index entries for sequences related to Lyndon words.

Cf. A000031, A001037, A028723, A051168. a(n) = T(n,4), array T as in A051168.
Cf. A000094.
Cf. A171238. - Gary W. Adamson, Dec 05 2009
Row sums of A173997. - Gary W. Adamson, Mar 05 2010
Column k=2 of A242093. Column k=2 of A115720 and A115994.
Cf. A117485, A257990, A307373, A325164, A325168.

a006918 n = a006918_list !! n
a006918_list = scanl (+) 0 a008805_list
-- Reinhard Zumkeller, Feb 01 2013

[Floor(Binomial(n+4, 4)/(n+4))-Floor((n+2)/8)*(1+(-1)^n)/2: n in [0..60]]; // Vincenzo Librandi, Nov 10 2014

with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=r),U=Sequence(Z,card>=3)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,ZL),size=m),m=11..58) ; # Zerinvary Lajos, Mar 09 2007
A006918 := proc(n)
    if type(n,'even') then
        n*(n+2)*(n+4)/24 ;
    else
        binomial(n+3,3)/4 ;
    fi ;
end proc: # R. J. Mathar, May 17 2016

f[n_]:=If[EvenQ[n],(n(n+2)(n+4))/24,Binomial[n+3,3]/4]; Join[{0},Array[f,60]]  (* Harvey P. Dale, Apr 20 2011 *)
durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]==2&]],{n,0,30}] (* Gus Wiseman, Apr 06 2019 *)

{ parttrees(n)=local(pt,k,nk); if (n%2==0, pt=(n/2+1)^2, pt=ceil(n/2)*(ceil(n/2)+1)); pt+=floor(n/2); for (x=1,floor(n/2),pt+=floor(x/2)+floor((n-x)/2)); if (n%2==0 && n>2, pt-=floor(n/4)); k=1; while (3*k<=n, for (x=k,floor((n-k)/2), pt+=floor(k/2); if (x!=k, pt+=floor(x/2)); if ((n-x-k)!=k && (n-x-k)!=x, pt+=floor((n-x-k)/2))); k++); pt }

{a(n) = n += 2; (n^3 - n * (2-n%2)^2) / 24}; /* Michael Somos, Aug 15 2009 */

G.f.: x/((1-x)^2*(1-x^2)^2) = x/((1+x)^2*(1-x)^4).
0, 0, 0, 1, 2, 5, 8, 14, ... has a(n) = (Sum_{k=0..n} floor(k(n-k)/2))/2. - Paul Barry, Sep 14 2003
0, 0, 0, 0, 0, 1, 2, 5, 8, 14, 20, 30, 40, 55, ... has a(n) = binomial(floor(1/2 n), 3) + binomial(floor(1/2 n + 1/2), 3) [Eke]. - N. J. A. Sloane, May 12 2012
a(0)=0, a(1)=1, a(n) = (2/(n-1))*a(n-1) + ((n+3)/(n-1))*a(n-2). - Benoit Cloitre, Jun 28 2004
a(n) = floor(binomial(n+4, 4)/(n+4)) - floor((n+2)/8)(1+(-1)^n)/2. - Paul Barry, Jan 01 2005
a(n+1) = a(n) + binomial(floor(n/2)+2,2), i.e., first differences are A008805. Convolution of A008619 with itself, then shifted right (or A004526 with itself, shifted left by 3). - Franklin T. Adams-Watters, Jan 27 2006
a(n+1) = (A027656(n) + A003451(n+5))/2 with a(1)=0. - Yosu Yurramendi, Sep 12 2008
Linear recurrence: a(n) = 2a(n-1) + a(n-2) - 4a(n-3) + a(n-4) + 2a(n-5) - a(n-6). - Jaume Oliver Lafont, Dec 05 2008
Euler transform of length 2 sequence [2, 2]. - Michael Somos, Aug 15 2009
a(n) = -a(-4-n) for all n in Z.
a(n+1) + a(n) = A002623(n). - Johannes W. Meijer, May 20 2011
a(n) = (n+2)*(2*n*(n+4)-3*(-1)^n+3)/48. - Bruno Berselli, May 21 2011
a(2n) = A007290(n+2). - Jon Perry, Nov 10 2014
G.f.: (1/(1-x)^4-1/(1-x^2)^2)/4. - Herbert Kociemba, Oct 23 2016
E.g.f.: (x*(18 + 9*x + x^2)*cosh(x) + (6 + 15*x + 9*x^2 + x^3)*sinh(x))/24. - Stefano Spezia, Dec 07 2021
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 75/4 - 24*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 69/4 - 24*log(2). (End)

A000241 Crossing number of complete graph with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315

Offset: 0

N. J. A. Sloane

It was conjectured by A. Hill in 1958 (see Guy 1960 and Harary-Hill 1963) that a(n) = floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2)/4 (see A028723). This is also sometimes referred to as Guy's conjecture. - N. J. A. Sloane, Jan 21 2015
Verified for n = 11, 12 by Shengjun Pan and R. Bruce Richter, in "The Crossing Number of K_11 is 100", J. Graph Theory 56 (2) (2007) 128-134.
Also the sum of the dimensions of the irreducible representations of su(3) that first occur in the (n-5)-th tensor power of the tautological representation. - James Dolan (jdolan(AT)math.ucr.edu), Jun 02 2003
From Paul Barry, Oct 02 2008: (Start)
Another version of the conjecture is that a(n)=C(floor(n/2),2)*C(floor((n-1)/2),2).
We conjecture that this sequence is also given by one half of the third coefficient of the denominator polynomial of the n-th convergent to the g.f. of n!.
(End)
From the Lackenby reference: "One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1#K2) = c(K1) + c(K2) always hold, where c(K) denotes the crossing number of a knot K and K1#K2 is the connected sum of two (oriented) knots K1 and K2? Theorem 1.1. Let K1, . . .,Kn be oriented knots in the 3-sphere. Then (c(K1) + . . . + c(Kn)) / 152 <= c(K1# . . . #Kn) <= c(K1) + . . . + c(Kn)." - Jonathan Vos Post, Aug 26 2009
From the Pan and Richter reference: 0.8594 Z(n) <= a(n) <= Z(n), where Z(n) is the conjectured formula (Richter and Thomassen 1997, de Klerk et al. 2007). - Danny Rorabaugh, Mar 12 2015
a(n) <= A028723(n) for n = 13-21, 23, 25, 27, and 29 based on crossing numbers equal to A028723(n) found using QuickCross. - Eric W. Weisstein, May 02 2019

Ábrego, Bernardo M.; Aichholzer, Oswin; Fernández-Merchant, Silvia; Ramos, Pedro; Salazar, Gelasio. The 2-Page Crossing Number of K_n. Discrete Comput. Geom. 49 (2013), no. 4, 747-777. MR3068573
E. de Klerk, D. V. Pasechnik, and A. Schrijver, "Reduction of Symmetric Semidefinite Programs Using the Regular *-Representation." Math Program. 109 (2007) 613-624.
Jean-Paul Delahaye, in Pour La Science, Feb. 2013, #424, Logique et Calcul. Le problème de la fabrique de briques. (The problem of the brick factory), in French.
R. K. Guy, The crossing number of the complete graph, Bull. Malayan Math. Soc., Vol. 7, pp. 68-72, 1960.
Harary, Frank, and Anthony Hill. "On the number of crossings in a complete graph." Proceedings of the Edinburgh Mathematical Society (Series 2) 13.04 (1963): 333-338.
T. L. Saaty, The number of intersections in complete graphs, Engrg. Cybernetics 9 (1971), no. 6, 1102-1104 (1972).; translated from Izv. Akad. Nauk SSSR Tehn. Kibernet. 1971, no. 6, 151-154 (Russian). Math. Rev. 58 #21749.
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).
C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.

Oswin Aichholzer, Another Small but Long Step for Crossing Numbers: cr(13) = 225 and cr(14) = 315, Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG 2021).
Martin Balko, Radoslav Fulek, and Jan Kynčl, Crossing numbers and combinatorial characterization of monotone drawings of K_n, arXiv preprint arXiv:1312.3679 [math.CO], 2013. Also Discrete Computat. Geom., 53 (2015), 107-143.
Drago Bokal, Gasper Fijavz and David R. Wood, The Minor Crossing Number of Graphs with an Excluded Minor, arXiv:math/0609707 [math.CO], 2006.
J. Dolan et al., su(3) and Zarankiewicz's conjecture
P. Erdős and R. K. Guy, Crossing Number Problems The American Mathematical Monthly, Vol. 80, No. 1. (1973), pp. 52-58.
James Grime and Brady Haran, The Brick Factory Problem, Numberphile video (2023).
Paul C. Kainen, On a problem of P. Erdős, J. Combinatorial Theory 51968 374--377. MR0231744 (38 #72)
Marc Lackenby, The crossing number of composite knots, arXiv:0805.4706 [math.GT], 2009.
D. McQuillan and R. B. Richter, A parity theorem for drawings of complete and bipartite graphs, Amer. Math. Monthly, 117 (2010), 267-273.
A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280.
A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280. [Annotated scanned copy]
Shengjun Pan and R. Bruce Richter, The Crossing Number of K_11 is 100, J. Graph Theory 56 (2) (2007) 128-134.
R. B. Richter and C. Thomassen, Relations Between Crossing Numbers of Complete and Complete Bipartite Graphs, Amer. Math. Monthly 104 (1997) 131-137.
Thomas L. Saaty, On polynomials and crossing numbers of complete graphs, J. Combinatorial Theory Ser. A 10 (1971), 183--184. MR0291013 (45 #107)
T. L. Saaty, The Minimum Number Of Intersections In Complete Graphs, PNAS 1964 52 (3) 688-690.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Graph Crossing Number
Eric Weisstein's World of Mathematics, Guy's Conjecture
Eric Weisstein's World of Mathematics, Zarankiewicz's Conjecture

It is conjectured that this sequence coincides with A028723.

Cf. A007333, A014540, A030179, A121021, A191928.

a(n) ~ n^4/64 (Guy, Kainen).

Empirical g.f.: -x^5*(1+x+x^2)/(x+1)^3/(x-1)^5, which is the same as the conjectured formula of A. Hill. - Simon Plouffe, Feb 06 2013

Bokal et al. link from Jonathan Vos Post, Dec 08 2006
Entry revised by N. J. A. Sloane, Jan 21 2015
Conjectured data values deleted by Eric W. Weisstein, May 01 2019
a(13) and a(14) computed by O. Aichholzer and added by Manfred Scheucher, Apr 24 2024

A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 9, 9, 3, 1, 1, 4, 12, 18, 18, 12, 4, 1, 1, 4, 16, 24, 36, 24, 16, 4, 1, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 6, 36, 90, 225, 300, 400, 300, 225, 90, 36, 6, 1

Offset: 1

Emeric Deutsch, Nov 24 2003

Rows 2, 4, 6, ... give A088459.
Diagonal sums are in A088518(n-1). - Philippe Deléham, Jan 04 2009
Row sums are in A001405(n). - Philippe Deléham, Jan 04 2009
Subtriangle (1 <= k <= n) of triangle T(n,k), 0 <= k <= n, read by rows, given by A101455 DELTA A056594 := [0,1,0,-1,0,1,0,-1,0,1,0,-1,0,...] DELTA [1,0,-1,0,1,0,-1,0,1,0,-1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
Also, number of symmetric noncrossing partitions of an n-set with k blocks. - Andrew Howroyd, Nov 15 2017
From Roger Ford, Oct 17 2018: (Start)
T(n,k) = t(n+2,d) where t(n,d) is the number of different semi-meander arch depth listings with n top arches and with d the depth of the deepest embedded arch.
Examples:    /\         semi-meander with 5 top arches
            //\\   /\   2 arches are at depth=0 (no covering arches)
           ///\\\ //\\  2 arches are at depth=1 (1 covering arch)
      (0)(1)(2)         1 arch is at depth=2 (2 covering arches)
       2, 2, 1 is the listing for this t(5,2)
             /\      semi-meander with 5 top arches
            /  \     (0)(1)
     /\ /\ //\/\\     3, 2  is the listing for this t(5,1)
a(6,5) = t(8,5)= 3 {2,1,1,1,2,1; 2,1,2,1,1,1; 3,1,1,1,1,1} (End)

			Triangle begins:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,   1;
  1,  2,  4,   2,   1;
  1,  3,  6,   6,   3,    1;
  1,  3,  9,   9,   9,    3,    1;
  1,  4, 12,  18,  18,   12,    4,    1;
  1,  4, 16,  24,  36,   24,   16,    4,    1;
  1,  5, 20,  40,  60,   60,   40,   20,    5,    1;
  1,  5, 25,  50, 100,  100,  100,   50,   25,    5,    1;
  1,  6, 30,  75, 150,  200,  200,  150,   75,   30,    6,   1;
  1,  6, 36,  90, 225,  300,  400,  300,  225,   90,   36,   6,   1;
  1,  7, 42, 126, 315,  525,  700,  700,  525,  315,  126,  42,   7,  1;
  1,  7, 49, 147, 441,  735, 1225, 1225, 1225,  735,  441, 147,  49,  7, 1;
  1,  8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1;
  ...
a(6,2)=3 because we have UUUDDDUUUDDD, UUUUDDUUDDDD, UUUUUDUDDDDD, where
U=(1,1), D=(1,-1).

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Hyunsoo Cho, JiSun Huh, and Jaebum Sohn, The (s, s + d, ..., s + pd)-core partitions and the rational Motzkin paths, arXiv:2001.06651 [math.CO], 2020.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Nicolas Crampe, Julien Gaboriaud, and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.
L. Poulain d'Andecy, Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics, arXiv:2304.00850 [math.RT], 2023. See p. 64.
Vladimir Shevelev, Several remarks on A088855, Seqfan thread, Nov 19 2017.

Cf. A005566, A018224, A056594, A084938, A088459, A101455, A209612, A247644.
Cf. A001405 (row sums), A088459, A088518 (diagonal sums).
Column 2 is A008619, column 3 is A002620, column 4 is A028724, column 5 is A028723, column 6 is A028725, column 7 is A331574.

[(&*[Binomial(Floor((n-j)/2), Floor((k-j)/2)): j in [0..1]]): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 08 2022

T[n_, k_] := Binomial[Quotient[n-1, 2], Quotient[k-1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]];
Table[T[n, k], {n,13}, {k,n}]//Flatten (* Jean-François Alcover, Jun 07 2018 *)

T(n,k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ Andrew Howroyd, Nov 15 2017

def A088855(n,k): return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
flatten([[A088855(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 08 2022

T(n, k) = binomial(floor(n'), floor(k'))*binomial(ceiling(n'), ceiling(k')), where n' = (n-1)/2, k' = (k-1)/2.
G.f.: 2*u/(u*v + sqrt(x*y*u*v)) - 1, where x = 1+z+t*z, y = 1+z-t*z, u = 1-z+t*z, v = 1-z-t*z.
Triangle T(n,k), 0 <= k <= n, given by A101455 DELTA A056594 begins: 1; 0,1; 0,1,1; 0,1,1,1; 0,1,2,2,1; 0,1,2,4,2,1; 0,1,3,6,6,3,1; 0,1,3,9,9,9,3,1; ... - Philippe Deléham, Jan 03 2009
From G. C. Greubel, Apr 08 2022: (Start)
T(n, n-k+1) = T(n, k).
T(2*n-1, n) = A018224(n-1), n >= 1.
T(2*n, n) = A005566(n-1), n >= 1. (End)

Keyword:tabl added Philippe Deléham, Jan 25 2010

A352666 Maximum number of induced copies of the claw graph K_{1,3} in an n-node graph.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 20, 40, 70, 112, 176, 261, 372, 520, 704, 935, 1220, 1560, 1976, 2464, 3038, 3710, 4480, 5376, 6392, 7548, 8856, 10320, 11970, 13800, 15840, 18095, 20580, 23320, 26312, 29601, 33176, 37072, 41300, 45875, 50830, 56160, 61920, 68096, 74732

Offset: 1

Pontus von Brömssen, Mar 26 2022

nonn

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to 1/2, the inducibility of the claw graph.

Brown and Sidorenko (1994) prove that a bipartite optimal graph (i.e., an n-node graph with a(n) induced claw graphs) exists for all n. For n >= 2, the size k of the smallest part of an optimal bipartite graph K_{k,n-k} is one of the two integers closest to n/2 - sqrt(3*n/4-1), and a(n) = binomial(k,3)*(n-k) + binomial(n-k,3)*k. Both are optimal if and only if n is in A271713. For 7 <= n <= 10 (and, trivially, n = 3), the tripartite graph K_{1,1,n-2} is also optimal.

Jason I. Brown and Alexander Sidorenko, The inducibility of complete bipartite graphs, Journal of Graph Theory 18 (1994), 629-645.

Cf. A271713.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352667 (paw graph), A352668 (diamond graph), A352669 (cycles).

from math import comb,isqrt
def A352666(n):
    if n <= 1: return 0
    r = isqrt(3*n-4)
    k0 = (n-r-1)//2
    return max(comb(k,3)*(n-k)+comb(n-k,3)*k for k in (k0,k0+1))

A352667 Maximum number of induced copies of the paw graph in an n-node graph.

Original entry on oeis.org

0, 0, 0, 1, 4, 9, 18, 36, 60, 97, 152, 224

Offset: 1

Pontus von Brömssen, Mar 26 2022

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to 3/8, the inducibility of the paw graph (Hirst 2014).
Assuming that the extremal graph is KK(j_1, j_2; k_1, k_2), as defined in the Example section below, for some j_1, j_2, k_1, k_2 (these graphs are asymptotically extremal), the sequence would continue as follows, with a(n) = (binomial(j_1,2)*j_2 + binomial(j_2,2)*j_1)*(k_1 + k_2) + (binomial(k_1,2)*k_2 + binomial(k_2,2)*k_1)*(j_1 + j_2).
   n |    a(n)     | (j_1, j_2), (k_1, k_2)
     |(conjectural)|
  -------------------------------------------------------
     316       (2,2), (4,5)
     440       (2,2), (5,5)
     590       (2,3), (5,5)
     780       (3,3), (5,5)
    1008       (2,3), (6,6), or (3,3), (5,6)
    1296       (3,3), (6,6)
    1620       (3,3), (6,7), or (3,4), (6,6)
    2016       (3,3), (7,7), or (4,4), (6,6)
    2478       (3,4), (7,7)
    3024       (4,4), (7,7)
    3632       (4,4), (7,8)
    4352       (4,4), (8,8)
    5152       (4,5), (8,8)
    6080       (5,5), (8,8)
    7100       (5,5), (8,9)
    8280       (5,5), (9,9)
    9558       (5,6), (9,9)
   11016       (6,6), (9,9)
   12600       (5,6), (10,10), or (6,6), (9,10)
   14400       (6,6), (10,10)
   16320       (6,6), (10,11), or (6,7), (10,10)
   18480       (6,6), (11,11), or (7,7), (10,10)
   20812       (6,7), (11,11)
   23408       (7,7), (11,11)
   26166       (7,7), (11,12)
   29232       (7,7), (12,12)
   32496       (7,8), (12,12)
   36096       (8,8), (12,12)
   39904       (8,8), (12,13)
   44096       (8,8), (13,13)
   48516       (8,9), (13,13)
   53352       (9,9), (13,13)
   58446       (9,9), (13,14)
   64008       (9,9), (14,14)
   69832       (9,10), (14,14)
   76160       (10,10), (14,14)
   82800       (9,10), (15,15), or (10,10), (14,15)
   90000       (10,10), (15,15)
For n > 10, more than one optimal graph of the form KK(j_1, j_2; k_1, k_2) seem to exist exactly when n = 2*m^2 + i, where m >= 3 and i = -1, 1, or 2.

			All extremal graphs (i.e., n-node graphs having a(n) induced paw graphs) for 4 <= n <= 12 are listed below. Here, KK(j_1, j_2; k_1, k_2) denotes the complement of the disjoint union of K_{j_1, j_2} and K_{k_1, k_2}.
  n = 4: KK(0,1;1,2) (the paw graph);
  n = 5: KK(0,1;2,2) (the butterfly graph);
  n = 6: KK(0,1;2,3);
  n = 7: KK(0,1;3,3), KK(0,2;2,3), and KK(1,1;2,3);
  n = 8: KK(0,2;3,3) and KK(1,1; 3,3);
  n = 9: KK(0,2;3,4), KK(1,1;3,4), and KK(1,2;3,3);
  n = 10: KK(1,2;3,4);
  n = 11: KK(1,2;4,4);
  n = 12: KK(2,2;4,4).

James Hirst, The inducibility of graphs on four vertices, Journal of Graph Theory 75 (2014), 231-243.
Falk Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 43e7869.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352666 (claw graph), A352668 (diamond graph), A352669 (cycles).

a(10)-a(12) added using tinygraph by Falk Hüffner, Apr 05 2022

A352668 Maximum number of induced copies of the diamond graph K_{1,1,2} in an n-node graph.

Original entry on oeis.org

0, 0, 0, 1, 4, 12, 24, 48, 84, 138, 216, 324, 459, 636, 864, 1152, 1488, 1900, 2400, 3000, 3675, 4470, 5400, 6480, 7668, 9030, 10584, 12348, 14259, 16408, 18816, 21504, 24384, 27576, 31104, 34992, 39123, 43650, 48600, 54000, 59700, 65890, 72600, 79860, 87483, 95742

Offset: 1

Pontus von Brömssen, Mar 26 2022

nonn

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to 72/125, the inducibility of the diamond graph (Hirst 2014).

It is known that there exists a complete multipartite optimal graph (i.e., an n-node graph with a(n) induced diamond graphs) for all n (Brown and Sidorenko 1994) and that a complete 5-partite graph is asymptotically optimal (Hirst 2014). For 3 <= n <= 7, the 3-partite Turán graph is optimal, and for 7 <= n <= 45, the 4-partite Turán graph is optimal. (For n = 7 both are optimal.) It appears that the 5-partite Turán graph is optimal for all n >= 46 (verified up to n = 75).

Jason I. Brown and Alexander Sidorenko, The inducibility of complete bipartite graphs, Journal of Graph Theory 18 (1994), 629-645.
James Hirst, The inducibility of graphs on four vertices, Journal of Graph Theory 75 (2014), 231-243.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352666 (claw graph), A352667 (paw graph), A352669 (cycles).

from sympy.utilities.iterables import combinations,partitions
from sympy.combinatorics import IntegerPartition
f=lambda p:sum(q[0]*q[1]*q[2]*(q[0]+q[1]+q[2]-3)//2 for q in combinations(p,3)) # number of induced diamond graphs in the multipartite graph with parts of sizes given by p
def A352668(n):
    return max(f(IntegerPartition(p).partition) for p in partitions(n))

A352669 Maximum number of induced cycles in an n-node graph.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 225

Offset: 1

Pontus von Brömssen, Mar 26 2022

For 3 <= n <= 11, a(n) = binomial(n,3) = A000292(n-2) and the complete graph is the unique extremal graph, but a(12) = 225 > binomial(12,3), where the unique extremal graph is K_{6,6}.

Morrison and Scott (2017) prove that, for sufficiently large n (they say it ought to be true for n >= 30), a(n) = A276401(n), with the unique extremal graph being the empty cyclic braid graph with one cluster of size 4 if n == 1 (mod 3), one cluster of size 2 if n == 2 (mod 3), and all other clusters of size 3. (The empty cyclic braid graph is obtained by arranging clusters of nodes of the appropriate sizes in a cycle and joining all pairs of nodes in neighboring clusters with edges.) For 14 <= n <= 21, this graph is not extremal, because the balanced bipartite graph K_{floor(n/2),ceiling(n/2)} has A028723(n+1) > A276401(n) induced cycles.

Falk Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 43e7869.
Natasha Morrison and Alex Scott, Maximising the number of induced cycles in a graph, Journal of Combinatorial Theory Series B 126 (2017), 24-61.

Cf. A000292, A028723, A276401.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352666 (claw graph), A352667 (paw graph), A352668 (diamond graph).

a(10)-a(12) added using tinygraph by Falk Hüffner, Apr 07 2022

A331574 a(n) is the number of subsets of {1..n} that contain 3 even and 3 odd numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 16, 40, 100, 200, 400, 700, 1225, 1960, 3136, 4704, 7056, 10080, 14400, 19800, 27225, 36300, 48400, 62920, 81796, 104104, 132496, 165620, 207025, 254800, 313600, 380800, 462400, 554880, 665856, 790704, 938961, 1104660, 1299600, 1516200, 1768900, 2048200, 2371600

Offset: 0

Enrique Navarrete, Jan 20 2020

			a(7) = 4 and the 4 subsets are {1,2,3,4,5,6}, {1,2,3,4,6,7}, {1,2,4,5,6,7}, {2,3,4,5,6,7}.

Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Cf. A028723 (2 even and 2 odd numbers).

[IsEven(n) select Binomial((n div 2),3)^2 else Binomial((n-1) div 2,3)*Binomial((n+1) div 2,3): n in [0..45]]; //  Marius A. Burtea, Jan 21 2020

a:= n-> ((b, q)-> b(q, 3)*b(n-q, 3))(binomial, iquo(n, 2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 30 2020

a[n_] := If[OddQ[n], Binomial[(n - 1)/2, 3]*Binomial[(n + 1)/2, 3], Binomial[n/2, 3]^2]; Array[a, 45, 0] (* Amiram Eldar, Jan 21 2020 *)
LinearRecurrence[{2,4,-10,-5,20,0,-20,5,10,-4,-2,1},{0,0,0,0,0,0,1,4,16,40,100,200},50] (* Harvey P. Dale, Dec 17 2022 *)

concat([0,0,0,0,0,0], Vec(x^6*(1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^7*(1 + x)^5) + O(x^40))) \\ Colin Barker, Jan 21 2020

a(n) = binomial(n/2,3)^2, n even;
a(n) = binomial((n-1)/2,3)*binomial((n+1)/2,3), n odd.
From Colin Barker, Jan 21 2020: (Start)
G.f.: x^6*(1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^7*(1 + x)^5).
a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n>11.
(End)
E.g.f.: (cosh(x)-sinh(x))*(45+36*x+18*x^2+6*x^3+3*x^4+(-45+54*x-36*x^2+18*x^3-9*x^4+6*x^5+2*x^6)*(cosh(2*x)+sinh(2*x)))/4608. - Stefano Spezia, Jan 27 2020

cp's OEIS Frontend

A342862 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 2143. 0 <= k <= A028723(n + 1).

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A342863 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1243. 0 <= k <= A028723(n + 1).

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A006918 a(n) = binomial(n+3, 3)/4 for odd n, n*(n+2)*(n+4)/24 for even n.

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A000241 Crossing number of complete graph with n nodes.

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A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.

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A352666 Maximum number of induced copies of the claw graph K_{1,3} in an n-node graph.

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Python

A352667 Maximum number of induced copies of the paw graph in an n-node graph.

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A352668 Maximum number of induced copies of the diamond graph K_{1,1,2} in an n-node graph.

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A006918 a(n) = binomial(n+3, 3)/4 for odd n, n(n+2)(n+4)/24 for even n.