A103905 -id:A103905 - cp's OEIS frontend

A002415 4-dimensional pyramidal numbers: a(n) = n^2*(n^2-1)/12.

0, 0, 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156, 58870, 67425, 76880, 87296, 98736, 111265, 124950, 139860, 156066, 173641, 192660, 213200, 235340

Offset: 0

N. J. A. Sloane

Also number of ways to legally insert two pairs of parentheses into a string of m := n-1 letters. (There are initially 2C(m+4,4) (A034827) ways to insert the parentheses, but we must subtract 2(m+1) for illegal clumps of 4 parentheses, 2m(m+1) for clumps of 3 parentheses, C(m+1,2) for 2 clumps of 2 parentheses and (m-1)C(m+1,2) for 1 clump of 2 parentheses, giving m(m+1)^2(m+2)/12 = n^2*(n^2-1)/12.) See also A000217.
E.g., for n=2 there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)), a((b)).
Let M_n denote the n X n matrix M_n(i,j)=(i+j); then the characteristic polynomial of M_n is x^(n-2) * (x^2-A002378(n)*x - a(n)). - Benoit Cloitre, Nov 09 2002
Let M_n denote the n X n matrix M_n(i,j)=(i-j); then the characteristic polynomial of M_n is x^n + a(n)x^(n-2). - Michael Somos, Nov 14 2002 [See A114327 for the infinite matrix M in triangular form. - Wolfdieter Lang, Feb 05 2018]
Number of permutations of [n] which avoid the pattern 132 and have exactly 2 descents. - Mike Zabrocki, Aug 26 2004
Number of tilings of a <2,n,2> hexagon.
a(n) is the number of squares of side length at least 1 having vertices at the points of an n X n unit grid of points (the vertices of an n-1 X n-1 chessboard). [For a proof, see Comments in A051602. - N. J. A. Sloane, Sep 29 2021] For example, on the 3 X 3 grid (the vertices of a 2 X 2 chessboard) there are four 1 X 1 squares, one (skew) sqrt(2) X sqrt(2) square, and one 3 X 3 square, so a(3)=6. On the 4 X 4 grid (the vertices of a 3 X 3 chessboard) there are 9 1 X 1 squares, 4 2 X 2 squares, 1 3 X 3 square, 4 sqrt(2) X sqrt(2) squares, and 2 sqrt(5) X sqrt(5) squares, so a(4) = 20. See also A024206, A108279. [Comment revised by N. J. A. Sloane, Feb 11 2015]
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005
Number of distinct components of the Riemann curvature tensor. - Gene Ward Smith, Apr 24 2006
a(n) is the number of 4 X 4 matrices (symmetrical about each diagonal) M = [a,b,c,d;b,e,f,c;c,f,e,b;d,c,b,a] with a+b+c+d=b+e+f+c=n+2; (a,b,c,d,e,f natural numbers). - Philippe Deléham, Apr 11 2007
If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-3) is the number of 5-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
a(n) is the number of Dyck (n+1)-paths with exactly n-1 peaks. - David Callan, Sep 20 2007
Starting (1,6,20,50,...) = third partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} C(n+3,i+3)*b(i), where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
4-dimensional square numbers. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
Equals row sums of triangle A177877; a(n), n > 1 = (n-1) terms in (1,2,3,...) dot (...,3,2,1) with additive carryovers. Example: a(4) = 20 = (1,2,3) dot (3,2,1) with carryovers = (1*3) + (2*2 + 3) + (3*1 + 7) = (3 + 7 + 10).
Convolution of the triangular numbers A000217 with the odd numbers A004273.
a(n+2) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and w-x=max{w,x,y,z}-min{w,x,y,z}. - Clark Kimberling, May 28 2012
The second level of finite differences is a(n+2) - 2*a(n+1) + a(n) = (n+1)^2, the squares. - J. M. Bergot, May 29 2012
Because the differences of this sequence give A000330, this is also the number of squares in an n+1 X n+1 grid whose sides are not parallel to the axes.
a(n+2) gives the number of 2*2 arrays that can be populated with 0..n such that rows and columns are nondecreasing. - Jon Perry, Mar 30 2013
For n consecutive numbers 1,2,3,...,n, the sum of all ways of adding the k-tuples of consecutive numbers for n=a(n+1). As an example, let n=4: (1)+(2)+(3)+(4)=10; (1+2)+(2+3)+(3+4)=15; (1+2+3)+(2+3+4)=15; (1+2+3+4)=10 and the sum of these is 50=a(4+1)=a(5). - J. M. Bergot, Apr 19 2013
If P(n,k) = n*(n+1)*(k*n-k+3)/6 is the n-th (k+2)-gonal pyramidal number, then a(n) = P(n,k)*P(n-1,k-1) - P(n-1,k)*P(n,k-1). - Bruno Berselli, Feb 18 2014
For n > 1, a(n) = 1/6 of the area of the trapezoid created by the points (n,n+1), (n+1,n), (1,n^2+n), (n^2+n,1). - J. M. Bergot, May 14 2014
For n > 3, a(n) is twice the area of a triangle with vertices at points (C(n,4),C(n+1,4)), (C(n+1,4),C(n+2,4)), and (C(n+2,4),C(n+3,4)). - J. M. Bergot, Jun 03 2014
a(n) is the dimension of the space of metric curvature tensors (those having the symmetries of the Riemann curvature tensor of a metric) on an n-dimensional real vector space. - Daniel J. F. Fox, Dec 15 2018
Coefficients in the terminating series identity 1 - 6*n/(n + 5) + 20*n*(n - 1)/((n + 5)*(n + 6)) - 50*n*(n - 1)*(n - 2)/((n + 5)*(n + 6)*(n + 7)) + ... = 0 for n = 1,2,3,.... Cf. A000330 and A005585. - Peter Bala, Feb 18 2019

			a(7) = 6*21 - (6*0 + 4*1 + 2*3 + 0*6 - 2*10 - 4*15) = 196. - _Bruno Berselli_, Jun 22 2013
G.f. = x^2 + 6*x^3 + 20*x^4 + 50*x^5 + 105*x^6 + 196*x^7 + 336*x^8 + ...

O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).
R. Euler and J. Sadek, "The Number of Squares on a Geoboard", Journal of Recreational Mathematics, 251-5 30(4) 1999-2000 Baywood Pub. NY
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]
Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
Henry Bottomley, Illustration of initial terms
Duane DeTemple, Using Squares to Sum Squares, The College Mathematics Journal, ? (2010), 214-221.
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
Reinhard O. W. Franz, and Berton A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), no. 1, 7-17.
M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052 [math.CO], 2012.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
M. Jones, S. Kitaev, and J. Remmel, Frame patterns in n-cycles, arXiv preprint arXiv:1311.3332 [math.CO], 2013.
Sandi Klavžar, Balázs Patkós, Gregor Rus, and Ismael G. Yero, On general position sets in Cartesian grids, arXiv:1907.04535 [math.CO], 2019.
G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31.
G. Kreweras, Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31. [Annotated scanned copy]
Calvin Lin, Squares on a grid, April 2015
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 9, 13, 15, 25.
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698).
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7
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
P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138. See Table I.
Royce A. Speck, The Number of Squares on a Geoboard, School Science and Mathematics, Volume 79, Issue 2, pages 145-150, February 1979
Eric Weisstein's World of Mathematics, Square Tiling.
Eric Weisstein's World of Mathematics, Riemann Tensor.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

a(n) = ((-1)^n)*A053120(2*n, 4)/8 (one-eighth of fifth unsigned column of Chebyshev T-triangle, zeros omitted). Cf. A001296.
Second row of array A103905.
Third column of Narayana numbers A001263.
Cf. A001079, A006011, A008911, A037270, A047819, A047928, A071253, A083374, A107891, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A108279, A024206.
Partial sums of A000330.
The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288, A134289, A134290, A134291, A140925, A140935, A169937.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers (A000027) with the k-gonal numbers.
Cf. A002388, A051602, A114327.

List([0..45],n->Binomial(n^2,2)/6); # Muniru A Asiru, Dec 15 2018

[n^2*(n^2-1)/12: n in [0..50]]; // Wesley Ivan Hurt, May 14 2014

A002415 := proc(n) binomial(n^2,2)/6 ; end proc: # Zerinvary Lajos, Jan 07 2008

Table[(n^4 - n^2)/12, {n, 0, 40}] (* Zerinvary Lajos, Mar 21 2007 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,1,6,20},40] (* Harvey P. Dale, Nov 29 2011 *)

PARI
```
a(n) = n^2 * (n^2 - 1) / 12;
```

x='x+O('x^200); concat([0, 0], Vec(x^2*(1+x)/(1-x)^5)) \\ Altug Alkan, Mar 23 2016

G.f.: x^2*(1+x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation
a(n) = Sum_{i=0..n} (n-i)*i^2 = a(n-1) + A000330(n-1) = A000217(n)*A000292(n-2)/n = A000217(n)*A000217(n-1)/3 = A006011(n-1)/3, convolution of the natural numbers with the squares. - Henry Bottomley, Oct 19 2000
a(n)+1 = A079034(n). - Mario Catalani (mario.catalani(AT)unito.it), Feb 12 2003
a(n) = 2*C(n+2, 4) - C(n+1, 3). - Paul Barry, Mar 04 2003
a(n) = C(n+2, 4) + C(n+1, 4). - Paul Barry, Mar 13 2003
a(n) = Sum_{k=1..n} A000330(n-1). - Benoit Cloitre, Jun 15 2003
a(n) = n*C(n+1,3)/2 = C(n+1,3)*C(n+1,2)/(n+1). - Mitch Harris, Jul 06 2006
a(n) = A006011(n)/3 = A008911(n)/2 = A047928(n-1)/12 = A083374(n)/6. - Zerinvary Lajos, May 09 2007
a(n) = (1/2)*Sum_{1 <= x_1, x_2 <= n} (det V(x_1,x_2))^2 = (1/2)*Sum_{1 <= i,j <= n} (i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007
a(n) = C(n+1,3) + 2*C(n+1,4). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = (1/48)*sinh(2*arccosh(n))^2. - Artur Jasinski, Feb 10 2010
a(n) = n*A000292(n-1)/2. - Tom Copeland, Sep 13 2011
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - Harvey P. Dale, Nov 29 2011
a(n) = (n-1)*A000217(n-1) - Sum_{i=0..n-2} (n-1-2*i)*A000217(i) for n > 1. - Bruno Berselli, Jun 22 2013
a(n) = C(n,2)*C(n+1,3) - C(n,3)*C(n+1,2). - J. M. Bergot, Sep 17 2013
a(n) = Sum_{k=1..n} ( (2k-n)* k(k+1)/2 ). - Wesley Ivan Hurt, Sep 26 2013
a(n) = floor(n^2/3) + 3*Sum_{k=1..n} k^2*floor((n-k+1)/3). - Mircea Merca, Feb 06 2014
Euler transform of length 2 sequence [6, -1]. - Michael Somos, May 28 2014
G.f. x^2*2F1(3,4;2;x). - R. J. Mathar, Aug 09 2015
Sum_{n>=2} 1/a(n) = 21 - 2*Pi^2 = 1.260791197821282762331... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A080852(2,n-2). - R. J. Mathar, Jul 28 2016
a(n) = A046092(n) * A046092(n-1)/48 = A000217(n) * A000217(n-1)/3. - Bruce J. Nicholson, Jun 06 2017
E.g.f.: (1/12)*exp(x)*x^2*(6 + 6*x + x^2). - Stefano Spezia, Dec 07 2018
Sum_{n>=2} (-1)^n/a(n) = Pi^2 - 9 (See A002388). - Amiram Eldar, Jun 28 2020

Typo in link fixed by Matthew Vandermast, Nov 22 2010

Redundant comment deleted and more detail on relationship with A000330 added by Joshua Zucker, Jan 01 2013

A008793 The problem of the calissons: number of ways to tile a hexagon of edge n with diamonds of side 1. Also number of plane partitions whose Young diagrams fit inside an n X n X n box.

Original entry on oeis.org

1, 2, 20, 980, 232848, 267227532, 1478619421136, 39405996318420160, 5055160684040254910720, 3120344782196754906063540800, 9265037718181937012241727284450000, 132307448895406086706107959899799334375000

Offset: 0

Jonas Wallgren

The 3-dimensional analog of A000984. - William Entriken, Aug 06 2013
The largest prime factor of a(n) is the largest prime p < 3*n. Its multiplicity is equal to 3*n-p. This can be proved with the formula of Michel Marcus, for example. - Walter Trump, Feb 11 2023
a(n) is also the number of resonance structures of circumcircum...coronene, where circum is repeated n-2 times where a(1) is the number of resonance structures of benzene (see Gutman et al.). - Yuan Yao, Oct 29 2023

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545, also p. 575 line -1 with a=b=c=n.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198. The first printing of Eq. (6.8) is wrong (see A049505 and A005157), but if one changes the limits in the formula (before it is corrected) to {1 <= i <= r, 1 <= j <= r}, one obtains the present sequence. - N. J. A. Sloane, Jun 30 2013
Gordon G. Cash and Jerry Ray Dias, Computation, Properties and Resonance Topology of Benzenoid Monoradicals and Polyradicals and the Eigenvectors Belonging to Their Zero Eigenvalues, J. Math. Chem., 30 (2001), 429-444. [See K, p. 442.]
Sebastien Desreux, Martin Matamala, Ivan Rapaport, Eric Remila, Domino tilings and related models: space of configurations of domains with holes, arXiv:math/0302344, 27 Feb 2003
Anne S. Meeussen, Erdal C. Oguz, Yair Shokef, Martin van Hecke1, Topological defects produce exotic mechanics in complex metamaterials, arXiv preprint 1903.07919, 2019 [See Section "Compatible metamaterials with fully antiferromagnetic interactions" - N. J. A. Sloane, Mar 23 2019]
J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see p. 261).

Seiichi Manyama, Table of n, a(n) for n = 0..54 (terms 0..30 from T. D. Noe)
T. Amdeberhan and V. H. Moll, Arithmetic properties of plane partitions, El. J. Comb. 18 (2) (2011) # P1.
Guy David and Carlos Tomei, The Problem of the Calissons, The American Mathematical Monthly, Vol. 96, No. 5 (May, 1989), pp. 429-431 (3 pages).
P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops, arXiv:math-ph/0410002, 2004.
I. Fischer, Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the center, arXiv:math/9906102 [math.CO], 1999.
P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.
M. Fulmek and C. Krattenthaler, The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II, arXiv:math/9909038 [math.CO], 1999.
I. Gutman, S. J. Cyvin, and V. Ivanov-Petrovic, Topological properties of circumcoronenes, Z. Naturforsch., 53a, 1998, 699-703 (see p. 700) - _Emeric Deutsch_, May 14 2018
H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
Sam Hopkins and Tri Lai, Plane partitions of shifted double staircase shape, arXiv:2007.05381 [math.CO], 2020. See Table 1 p. 9.
C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507v2 [math.CO], 2005.
P. A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.
Anne S. Meeussen, Erdal C. Oguz, Yair Shokef, and Martin van Hecke, Topological defects produce exotic mechanics in complex metamaterials, arXiv:1903.07919 [cond-mat.soft], 2019.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
James Propp, Updated article
James Propp, Tiling Problems, Old and New, Rutgers University Math Colloquium, March 30, 2022
N. C. Saldanha and C. Tomei, An overview of domino and lozenge tilings, arXiv:math/9801111 [math.CO], 1998.
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
Walter Trump, Prime factorization of a(n) for 1..950
Eric Weisstein's World of Mathematics, Plane Partition.

Cf. A000984, A066931, A352656, A352657. Main diagonal of array A103905.

A008793 := proc(n) local i; mul((i - 1)!*(i + 2*n - 1)!/((i + n - 1)!)^2, i = 1 .. n) end proc;

Table[ Product[ (i+j+k-1)/(i+j+k-2), {i, n}, {j, n}, {k, n} ], {n, 10} ]

a(n) = prod(i=1,n, prod(j=1, n, (n+i+j-1)/(i+j-1))); \\ Michel Marcus, Jul 13 2020

a(n) = Product_{i = 0..n-1} (i^(-i)*(n+i)^(2*i-n)*(2*n+i)^(n-i)).
a(n) = Product_{i = 1..n} Product_{j = 0..n-1} (3*n-i-j)/(2*n-i-j).
a(n) = Product_{i = 1..n} Gamma[i]*Gamma[i+2*n]/Gamma[i+n]^2.
a(n) = Product_{i = 0..n-1} i!*(i+2*n)!/(i+n)!^2.
a(n) = Product_{i = 1..n} Product_{j = n..2*n-1} i+j / Product_{j = 0..n-1} i+j. - Paul Barry, Jun 13 2006
For n >= 1, a(n) = det(binomial(2*n,n+i-j)) for 1<=i,j<=n [Krattenhaller, Theorem 4, with a = b = c = n].
Let H(n) = Product_{k = 1..n-1} k!. Then for a,b,c nonnegative integers (H(a)*H(b)*H(c)*H(a+b+c))/(H(a+b)*H(b+c)*H(c+a)) is an integer [MacMahon, Chapter II, Section 429, p. 182, with x -> 1]. Setting a = b = c = n gives the entries for this sequence. - Peter Bala, Dec 22 2011
a(n) ~ exp(1/12) * 3^(9*n^2/2 - 1/12) / (A * n^(1/12) * 2^(6*n^2 - 1/4)), where A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 27 2015
a(n) = Product_{i = 1..n} Product_{j = 1..n} (n+i+j-1)/(i+j-1). - Michel Marcus, Jul 13 2020
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1))^p (mod p^(4*r)) hold for all primes p and positive integers n and r. - Peter Bala, Apr 07 2022

More terms from Eric W. Weisstein

A047819 a(n) = Product_{i=1..n} ((i+3)(i+4)(i+5))/(i(i+1)(i+2)).

Original entry on oeis.org

1, 20, 175, 980, 4116, 14112, 41580, 108900, 259545, 572572, 1184183, 2318680, 4331600, 7768320, 13441968, 22535064, 36729945, 58373700, 90684055, 138003404, 206108980, 302588000, 437287500, 622849500, 875343105, 1215006156, 1667110095

Offset: 0

N. J. A. Sloane

Number of tilings of a <3,n,3> hexagon.
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 18 2005
Partial sums of A107891. - Peter Bala, Sep 21 2007
Determinant of the 3 X 3 matrix with rows [C(n+3,3) C(n+3,4) C(n+3,5)], [C(n+4,3) C(n+4,4) C(n+4,5)], and [C(n+5,3) C(n+5,4) C(n+5,5)]. - J. M. Bergot, Sep 10 2013

			G.f. = 1 + 20*x + 175*x^2 + 980*x^3 + 4116*x^4 + 14112*x^5 + 41580*x^6 + ...

O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 232, # 2 and p. 105, eq.(ii), K(0a(2,5,n))).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Paolo Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.2(ii), case a=2].
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 25.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Third row of array A103905.

Cf. A002415, A107891, A107915, A133708.

a:=n->(n+1)*(n+2)^2*(n+3)^3*(n+4)^2*(n+5)/8640: seq(a(n),n=0..30); # Emeric Deutsch, Jun 18 2005

a[n_] :=(n + 1)*(n + 2)^2*(n + 3)^3*(n + 4)^2*(n + 5)/8640;
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 19 2018, after Emeric Deutsch *)

{a(n) = if( n<0, 0, binomial(n+5, 5) * binomial(n+4, 3) * (n+3) / 12)}; /* Michael Somos, Nov 14 2002 */

{a(n) = my(s=sign(n+3)); n=abs(n+3)-3; -s/8 * polcoeff( charpoly( matrix(n+3, n+3, i, j, (i-j)^2)), n)}; /* Michael Somos, Nov 14 2002 */

G.f.: (1 + 10*x + 20*x^2 + 10*x^3 + x^4) / (1 - x)^10. - Michael Somos, Nov 14 2002
a(n) = C(n+3,n+2)*C(n+4,n+1)*C(n+5,n)/12. - Zerinvary Lajos, May 29 2007
a(n-3) = (1/24)*Sum_{1 <= x_1, x_2, x_3 <= n} (det V(x_1,x_2,x_3))^2 = (1/24)*Sum_ {1 <= i,j,k <= n} ((i-j)(i-k)(j-k))^2, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. - Peter Bala, Sep 21 2007
a(n) = -a(-6-n) for all n in Z. - Michael Somos, Dec 26 2016
From Amiram Eldar, May 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 5195/2 - 2160*zeta(3).
Sum_{n>=0} (-1)^n/a(n) = 17205/2 - 9600*log(2) - 1620*zeta(3). (End)

A120258 Triangle of central coefficients of generalized Pascal-Narayana triangles.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 20, 4, 1, 1, 70, 175, 50, 5, 1, 1, 252, 1764, 980, 105, 6, 1, 1, 924, 19404, 24696, 4116, 196, 7, 1, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1

Offset: 0

Paul Barry, Jun 13 2006

Columns are the central coefficients of the triangles T(n, k;r) with T(n, k;r)=Product{j=0..r, C(n+j, k+j)/C(n-k+j, j)}*[k<=n]; (r=0,A007318), (r=1;A001263),(r=2,A056939),(r=3,A056940),(r=4,A056941). Essentially A103905 as a number triangle with an extra diagonal of 1's. Central coefficients T(2n, n) are A008793. Row sums are A120259. Diagonal sums are A120260.

			Triangle begins:
  1;
  1,   1;
  1,   2,     1;
  1,   6,     3,     1;
  1,  20,    20,     4,    1;
  1,  70,   175,    50,    5,   1;
  1, 252,  1764,   980,  105,   6, 1;
  1, 924, 19404, 24696, 4116, 196, 7, 1;
  ...

Seiichi Manyama, Rows n = 0..100, flattened

Row sums give A120259.

Cf. A000891, A000984, A103905, A120257.

T(n, k) = prod(j=0, k-1, binomial(2*n-2*k+j, n-k)/binomial(n-k+j, j)); \\ Seiichi Manyama, Apr 02 2021

Number triangle T(n, k)=[k<=n]*Product{j=0..k-1, C(2n-2k+j, n-k)/C(n-k+j, j)}

As a square array, this is T(n,m)=product{k=1..m, product{j=1..n, product{i=1..n, (i+j+k-1)/(i+j+k-2)}}}; - Paul Barry, May 13 2008

A133112 Triangular array, read by rows, associated with sums of certain Vandermonde determinants.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 10, 8, 1, 5, 20, 35, 16, 1, 6, 35, 112, 126, 32, 1, 7, 56, 294, 672, 462, 64, 1, 8, 84, 672, 2772, 4224, 1716, 128, 1, 9, 120, 1386, 9504, 28314, 27456, 6435, 256, 1, 10, 165, 2640, 28314, 151008, 306735, 183040, 24310, 512, 1

Offset: 1

Peter Bala, Sep 18 2007

Appears to be the same as A073165 read as a triangular array (excluding the first column).

			Triangle starts:
  1;
  2 1;
  3 4 1;
  4 10 8 1;
  5 20 35 16 1;

A000292 (column 2), A040977 (column 3), A133111 (column 4). Cf. A103905.

T(n,k)=1/(1!*2! ... *k!)*Sum_{1 <= x_1, ...,x_k <= n} |det V(x_1, ...,x_k)|, where V(x_1, ...,x_k) is the Vandermonde matrix of order k. For example, T(n,2) = 1/2*Sum_{1<=i,j<= n} |i-j|. T(n,k) = 1/(1!*2! ... *k!)*Sum_{1 <= x_1, ...,x_k <= n} |(Product_{1 <= i < j <= k} (x_j - x_i) )|.

A047835 a(n) = Product_{i=1..n} ((i+4)(i+5)(i+6)(i+7))/(i(i+1)(i+2)(i+3)).

Original entry on oeis.org

1, 70, 1764, 24696, 232848, 1646568, 9343620, 44537922, 184225041, 677352676, 2254684432, 6892441920, 19571505408, 52101067968, 131018862096, 313203587004, 715536058545, 1569305708586, 3316911815140, 6778924352200, 13435361082000

Offset: 0

N. J. A. Sloane

nonn

Number of tilings of a <4,n,4> hexagon.

Partial sums of A133708. - Peter Bala, Sep 21 2007

O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.

T. D. Noe, Table of n, a(n) for n = 0..1000
O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 25.

Fourth row of array A103905.

Cf. A002415, A047819, A133708.

seq(binomial(n,n-1)*binomial(n+1,n-2)*binomial(n+2,n-3)*binomial(n+3,n-4)/(10*4!), n=4..24); # Zerinvary Lajos, May 29 2007

Table[Product[Times@@((i+Range[4,7])/(i+Range[0,3])),{i,n}],{n,0,20}] (* Harvey P. Dale, Nov 03 2011 *)

a(n) = C(n,n-1)*C(n+1,n-2)*C(n+2,n-3)*C(n+3,n-4)/(10*4!), n >= 4 . - Zerinvary Lajos, May 29 2007
a(n-4) = (1/3456)*Sum_{1 <= x_1, x_2, x_3, x_4 <= n} (det V(x_1,x_2,x_3,x_4))^2 = (1/3456)*Sum_{1 <= i,j,k,l <= n} ((i-j)(i-k)(i-l)(j-k)(j-l)(k-l))^2, where V(x_1,x_2,x_3,x_4) is the Vandermonde matrix of order 4. - Peter Bala, Sep 21 2007
Empirical g.f.: (x+1)*(x^8 + 52*x^7 + 658*x^6 + 2890*x^5 + 4810*x^4 + 2890*x^3 + 658*x^2 + 52*x + 1)/(1-x)^17. - Colin Barker, Jun 06 2012
Sum_{n>=0} 1/a(n) = 67200*Pi^4 + 5605600*Pi^2 - 185612833/3. - Amiram Eldar, May 29 2022

A071095 Number of ways to tile hexagon of edges n, n+1, n+1, n, n+1, n+1 with diamonds of side 1.

Original entry on oeis.org

1, 6, 175, 24696, 16818516, 55197331332, 872299918503728, 66345156372852988800, 24277282058281388285162560, 42730166102274086598901662210000, 361690697335823816369045433734882109375, 14721491647169381835282394824891766183125000000, 2880942480871157389699990094736740229925045312500000000

Offset: 0

N. J. A. Sloane, May 28 2002

nonn

J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see page 261).

Seiichi Manyama, Table of n, a(n) for n = 0..53
J. Propp, Updated article
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

Cf. A008793, A103905.

Table[Product[(i+j+k+2)/(i+j+k+1),{i,0,n-1},{j,0,n},{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Apr 26 2015 *)

a(n) = prod(i=0, n-1, prod(j=0, n, prod(k=0, n, (i+j+k+2)/(i+j+k+1)))) \\ Michel Marcus, May 20 2013

a(n) = Product_{i=0..a-1} Product_{j=0..b-1} Product_{k=0..c-1} (i+j+k+2)/(i+j+k+1) with a=n, b=c=n+1.
a(n) ~ exp(1/12) * 3^(9*n^2/2 + 6*n + 23/12) / (A * n^(1/12) * 2^(6*n^2 + 8*n + 11/4)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 26 2015
a(n) = (-1)^floor(n/2)*det(M(n)) where M(n) is the n X n matrix with m(i,j) = binomial(2*n+i+j,i+j). - Benoit Cloitre, Oct 22 2022

A000819 E.g.f.: cos(x)^2 / cos(2x) = Sum_{n >= 0} a(n) * x^(2n) / (2n)!.

Original entry on oeis.org

1, 2, 40, 1952, 177280, 25866752, 5535262720, 1633165156352, 635421069967360, 315212388819402752, 194181169538675507200, 145435130631317935357952, 130145345400688287667978240, 137139396592145493713802493952

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 2*x + 40*x^2 + 1952*x^3 + 177280*x^4 + 25866752*x^5 + ... - _Michael Somos_, Apr 04 2017

R. J. Mathar, Table of n, a(n) for n = 0..200

Essentially the same as A000816.
Second column of array A103905.
Cf. A000364, A002436.

With[{nn=30},Take[CoefficientList[Series[Cos[x]^2/Cos[2x],{x,0,nn}],x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jul 06 2014 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ 1 / (1 - Tan[x]^2), {x, 0, m}]]]; (* Michael Somos, Apr 04 2017 *)

{a(n) = my(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / (1 - tan(x + x * O(x^m))^2), m))}; /* Michael Somos, Apr 04 2017 */

E.g.f.: cos(x)^2/cos(2x)=1/Q(0)+1/2;   Q(k)=1+1/(1-2*(x^2)/(2*(x^2)-(k+1)*(2k+1)/Q(k+1)));  (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011
From Michael Somos, Apr 04 2017: (Start)
E.g.f.: cos(x)^2 / cos(2*x) = (1 + sec(2*x)) / 2 = tan(2*x) / (2 * tan(x)) = 1 / (1 - tan(x)^2).
a(n) = A000816(n) unless n=0.
a(n) = 1/2 * A002436(n) unless n=0.
a(n) = 2^(2*n - 1) * A000364(n). (End)

A047831 a(n) = Product_{i=1..n} ((i+5)(i+6)(i+7)(i+8)(i+9))/(i(i+1)(i+2)(i+3)(i+4)).

Original entry on oeis.org

1, 252, 19404, 731808, 16818516, 267227532, 3184461423, 30107635272, 235234907908, 1566039386912, 9095857138368, 46960429261824, 218772384397632, 931020034054176, 3656383418054268, 13365232267026024, 45800747571406905, 148055097314224100

Offset: 0

N. J. A. Sloane

Number of tilings of a <5,n,5> hexagon.

O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 25.

Fifth row of array A103905.

Cf. A002117, A013663.

seq(binomial(n+5,n+4)*binomial(n+6,n+3)*binomial(n+7,n+2)*binomial(n+8,n+1)*binomial(n+9,n)/(140*5!), n=0..17); # Zerinvary Lajos, May 29 2007

Table[Product[Times@@(i+Range[5,9])/Times@@(i+Range[0,4]),{i,n}],{n,0,20}] (* Harvey P. Dale, Jan 30 2015 *)

a(n) = prod(k=1, 5, binomial(n+k+4, n-k+5))/(140*5!); \\ Seiichi Manyama, Apr 02 2021

a(n) = C(n+5,n+4)*C(n+6,n+3)*C(n+7,n+2)*C(n+8,n+1)*C(n+9,n)/(140*5!). - Zerinvary Lajos, May 29 2007

Sum_{n>=0} 1/a(n) = 876090600000*zeta(3) + 266716800000*zeta(5) - 5318707224821/4. - Amiram Eldar, Aug 31 2025

Definition corrected by Daniel Soll (soll(AT)mathematik.uni-marburg.de), Aug 31 2004

A071094 Number of ways to tile hexagon of edges n, n, n+1, n, n, n+1 with diamonds of side 1.

Original entry on oeis.org

1, 3, 50, 4116, 1646568, 3184461423, 29706808370096, 1335119245893326400, 288882990167192721013376, 300792059519113653077154558000, 1506680146887473588202049621593937500, 36298820709557430183399305000196605531250000, 4205446372314569673006362329181090368935937500000000, 2342761095072644391194625697884219372917666852341417500000000

Offset: 0

N. J. A. Sloane, May 28 2002

J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see page 261).

Seiichi Manyama, Table of n, a(n) for n = 0..53
J. Propp, Updated article
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

Cf. A008793, A071095, A103905.

Table[Product[(i+j+k+2)/(i+j+k+1),{i,0,n-1},{j,0,n-1},{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Apr 26 2015 *)

a(n) = prod(k=0, n, binomial(2*n+k,n+k)/binomial(n+k,k)) \\ Michel Marcus, May 20 2013

a(n) = Product_{i=0..a-1} Product_{j=0..b-1} Product_{k=0..c-1} (i+j+k+2)/(i+j+k+1) with a=b=n, c=n+1.
a(n) = Product_{k=0..n} C(2n+k,n+k)/C(n+k,k). - Paul Barry, May 13 2008
a(n) ~ exp(1/12) * 3^(9*n^2/2 + 3*n + 5/12) / (A * n^(1/12) * 2^(6*n^2 + 4*n + 3/4)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 26 2015

cp's OEIS Frontend

A002415 4-dimensional pyramidal numbers: a(n) = n^2*(n^2-1)/12.

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A008793 The problem of the calissons: number of ways to tile a hexagon of edge n with diamonds of side 1. Also number of plane partitions whose Young diagrams fit inside an n X n X n box.

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A047819 a(n) = Product_{i=1..n} ((i+3)*(i+4)*(i+5))/(i*(i+1)*(i+2)).

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A120258 Triangle of central coefficients of generalized Pascal-Narayana triangles.

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A133112 Triangular array, read by rows, associated with sums of certain Vandermonde determinants.

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A047835 a(n) = Product_{i=1..n} ((i+4)*(i+5)*(i+6)*(i+7))/(i*(i+1)*(i+2)*(i+3)).

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A071095 Number of ways to tile hexagon of edges n, n+1, n+1, n, n+1, n+1 with diamonds of side 1.

A047819 a(n) = Product_{i=1..n} ((i+3)(i+4)(i+5))/(i(i+1)(i+2)).

A047835 a(n) = Product_{i=1..n} ((i+4)(i+5)(i+6)(i+7))/(i(i+1)(i+2)(i+3)).

A047831 a(n) = Product_{i=1..n} ((i+5)(i+6)(i+7)(i+8)(i+9))/(i(i+1)(i+2)(i+3)(i+4)).