A002415 - OEIS

%I M4135 N1714 #342 Jan 07 2025 10:52:14

%S 0,0,1,6,20,50,105,196,336,540,825,1210,1716,2366,3185,4200,5440,6936,

%T 8721,10830,13300,16170,19481,23276,27600,32500,38025,44226,51156,

%U 58870,67425,76880,87296,98736,111265,124950,139860,156066,173641,192660,213200,235340

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

%C 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.

%C E.g., for n=2 there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)), a((b)).

%C 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

%C 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]

%C Number of permutations of [n] which avoid the pattern 132 and have exactly 2 descents. - _Mike Zabrocki_, Aug 26 2004

%C Number of tilings of a <2,n,2> hexagon.

%C 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]

%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 12 2005

%C Number of distinct components of the Riemann curvature tensor. - _Gene Ward Smith_, Apr 24 2006

%C 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

%C 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

%C a(n) is the number of Dyck (n+1)-paths with exactly n-1 peaks. - _David Callan_, Sep 20 2007

%C 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

%C 4-dimensional square numbers. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

%C 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).

%C Convolution of the triangular numbers A000217 with the odd numbers A004273.

%C 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

%C 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

%C 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.

%C 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

%C 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

%C 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

%C 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

%C 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

%C 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

%C 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

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

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.

%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).

%D 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

%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002415/b002415.txt">Table of n, a(n) for n = 0..1000</a>

%H P. Aluffi, <a href="http://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, 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]."]

%H O. D. Anderson, <a href="/A002415/a002415.pdf">Find the next sequence</a>, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]

%H Brandy Amanda Barnette, <a href="http://digitalcommons.wku.edu/theses/1484">Counting Convex Sets on Products of Totally Ordered Sets</a>, Masters Theses & Specialist Projects, Paper 1484, 2015.

%H Henry Bottomley, <a href="/A002415/a002415.gif">Illustration of initial terms</a>

%H Duane DeTemple, <a href="http://www.maa.org/publications/periodicals/college-mathematics-journal/college-mathematics-journal-contents-may-2010">Using Squares to Sum Squares</a>, The College Mathematics Journal, ? (2010), 214-221.

%H Steven Edwards and William Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Griffiths/griffiths51.html">On Generalized Delannoy Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.

%H Reinhard O. W. Franz, and Berton A. Earnshaw, <a href="http://dx.doi.org/10.1007/s00026-002-8026-z">A constructive enumeration of meanders</a>, Ann. Comb. 6 (2002), no. 1, 7-17.

%H M. Hyatt and J. Remmel, <a href="http://arxiv.org/abs/1208.1052">The classification of 231-avoiding permutations by descents and maximum drop</a>, arXiv preprint arXiv:1208.1052 [math.CO], 2012.

%H Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>

%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.

%H M. Jones, S. Kitaev, and J. Remmel, <a href="http://arxiv.org/abs/1311.3332">Frame patterns in n-cycles</a>, arXiv preprint arXiv:1311.3332 [math.CO], 2013.

%H Sandi Klavžar, Balázs Patkós, Gregor Rus, and Ismael G. Yero, <a href="https://arxiv.org/abs/1907.04535">On general position sets in Cartesian grids</a>, arXiv:1907.04535 [math.CO], 2019.

%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1967__10__23_0">Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb"</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31.

%H G. Kreweras, <a href="/A006542/a006542_1.pdf">Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb"</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31. [Annotated scanned copy]

%H Calvin Lin, <a href="https://brilliant.org/discussions/thread/squares-on-a-grid">Squares on a grid</a>, April 2015

%H Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See pp. 9, 13, 15, 25.

%H C. P. Neuman and D. I. Schonbach, <a href="http://dx.doi.org/10.1137/1019006">Evaluation of sums of convolved powers using Bernoulli numbers</a>, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698).

%H C. J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq. 16 (2013) #13.5.7

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H P. N. Rathie, <a href="http://dx.doi.org/10.1016/0095-8956(74)90055-0">A census of simple planar triangulations</a>, J. Combin. Theory, B 16 (1974), 134-138. See Table I.

%H Royce A. Speck, <a href="http://dx.doi.org/10.1111/j.1949-8594.1979.tb09466.x">The Number of Squares on a Geoboard</a>, School Science and Mathematics, Volume 79, Issue 2, pages 145-150, February 1979

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannTensor.html">Riemann Tensor.</a>

%H A. F. Y. Zhao, <a href="http://www.emis.de/journals/JIS/VOL17/Zhao/zhao3.html">Pattern Popularity in Multiply Restricted Permutations</a>, Journal of Integer Sequences, 17 (2014), #14.10.3.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F G.f.: x^2*(1+x)/(1-x)^5. - _Simon Plouffe_ in his 1992 dissertation

%F 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

%F a(n)+1 = A079034(n). - Mario Catalani (mario.catalani(AT)unito.it), Feb 12 2003

%F a(n) = 2*C(n+2, 4) - C(n+1, 3). - _Paul Barry_, Mar 04 2003

%F a(n) = C(n+2, 4) + C(n+1, 4). - _Paul Barry_, Mar 13 2003

%F a(n) = Sum_{k=1..n} A000330(n-1). - _Benoit Cloitre_, Jun 15 2003

%F a(n) = n*C(n+1,3)/2 = C(n+1,3)*C(n+1,2)/(n+1). - _Mitch Harris_, Jul 06 2006

%F a(n) = A006011(n)/3 = A008911(n)/2 = A047928(n-1)/12 = A083374(n)/6. - _Zerinvary Lajos_, May 09 2007

%F 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

%F a(n) = C(n+1,3) + 2*C(n+1,4). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

%F a(n) = (1/48)*sinh(2*arccosh(n))^2. - _Artur Jasinski_, Feb 10 2010

%F a(n) = n*A000292(n-1)/2. - _Tom Copeland_, Sep 13 2011

%F 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

%F 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

%F a(n) = C(n,2)*C(n+1,3) - C(n,3)*C(n+1,2). - _J. M. Bergot_, Sep 17 2013

%F a(n) = Sum_{k=1..n} ( (2k-n)* k(k+1)/2 ). - _Wesley Ivan Hurt_, Sep 26 2013

%F a(n) = floor(n^2/3) + 3*Sum_{k=1..n} k^2*floor((n-k+1)/3). - _Mircea Merca_, Feb 06 2014

%F Euler transform of length 2 sequence [6, -1]. - _Michael Somos_, May 28 2014

%F G.f. x^2*2F1(3,4;2;x). - _R. J. Mathar_, Aug 09 2015

%F Sum_{n>=2} 1/a(n) = 21 - 2*Pi^2 = 1.260791197821282762331... . - _Vaclav Kotesovec_, Apr 27 2016

%F a(n) = A080852(2,n-2). - _R. J. Mathar_, Jul 28 2016

%F a(n) = A046092(n) * A046092(n-1)/48 = A000217(n) * A000217(n-1)/3. - _Bruce J. Nicholson_, Jun 06 2017

%F E.g.f.: (1/12)*exp(x)*x^2*(6 + 6*x + x^2). - _Stefano Spezia_, Dec 07 2018

%F Sum_{n>=2} (-1)^n/a(n) = Pi^2 - 9 (See A002388). - _Amiram Eldar_, Jun 28 2020

%e a(7) = 6*21 - (6*0 + 4*1 + 2*3 + 0*6 - 2*10 - 4*15) = 196. - _Bruno Berselli_, Jun 22 2013

%e G.f. = x^2 + 6*x^3 + 20*x^4 + 50*x^5 + 105*x^6 + 196*x^7 + 336*x^8 + ...

%p A002415 := proc(n) binomial(n^2,2)/6 ; end proc: # _Zerinvary Lajos_, Jan 07 2008

%t Table[(n^4 - n^2)/12, {n, 0, 40}] (* _Zerinvary Lajos_, Mar 21 2007 *)

%t LinearRecurrence[{5,-10,10,-5,1},{0,0,1,6,20},40] (* _Harvey P. Dale_, Nov 29 2011 *)

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

%o (PARI) x='x+O('x^200); concat([0, 0], Vec(x^2*(1+x)/(1-x)^5)) \\ _Altug Alkan_, Mar 23 2016

%o (Magma) [n^2*(n^2-1)/12: n in [0..50]]; // _Wesley Ivan Hurt_, May 14 2014

%o (GAP) List([0..45],n->Binomial(n^2,2)/6); # _Muniru A Asiru_, Dec 15 2018

%Y a(n) = ((-1)^n)*A053120(2*n, 4)/8 (one-eighth of fifth unsigned column of Chebyshev T-triangle, zeros omitted). Cf. A001296.

%Y Second row of array A103905.

%Y Third column of Narayana numbers A001263.

%Y Cf. A001079, A006011, A008911, A037270, A047819, A047928, A071253, A083374, A107891, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A108279, A024206.

%Y Partial sums of A000330.

%Y 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.

%Y Cf. A220212 for a list of sequences produced by the convolution of the natural numbers (A000027) with the k-gonal numbers.

%Y Cf. A002388, A051602, A114327.

%K nonn,easy,nice

%O 0,4

%A _N. J. A. Sloane_

%E Typo in link fixed by _Matthew Vandermast_, Nov 22 2010

%E Redundant comment deleted and more detail on relationship with A000330 added by _Joshua Zucker_, Jan 01 2013