%I A002415 M4135 N1714
%S A002415 0,0,1,6,20,50,105,196,336,540,825,1210,1716,2366,3185,4200,5440,6936,
%T A002415 8721,10830,13300,16170,19481,23276,27600,32500,38025,44226,51156,
%U A002415 58870,67425,76880,87296,98736,111265,124950,139860,156066,173641
%N A002415 4-dimensional pyramidal numbers: n^2*(n^2-1)/12.
%C A002415 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 A002415 E.g. for n=2 there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)), 
               a((b)).
%C A002415 Let M_n denotes 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 (benoit7848c(AT)orange.fr), Nov 09 2002
%C A002415 Let M_n denotes 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
%C A002415 a(n)+1 is the determinant of the n X n matrix M with M(i,i)=1, M(i,j)=i-j. 
               - Mario Catalani (mario.catalani(AT)unito.it), Feb 12 2003
%C A002415 Number of permutations of [n] which avoid the pattern 132 and have exactly 
               2 descents. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 
               26 2004
%C A002415 Number of tilings of a <2,n,2> hexagon.
%C A002415 a(n) = number of squares with corners on an n X n grid. See also A024206, 
               A108279.
%C A002415 Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jun 12 2005
%C A002415 Number of distinct components of the Riemann curvature tensor. - Gene 
               Ward Smith (genewardsmith(AT)gmail.com), Apr 24 2006
%C A002415 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 DELEHAM (kolotoko(AT)wanadoo.fr), 
               Apr 11 2007
%C A002415 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 R. Janjic (agnus(AT)blic.net), Sep 19 2007
%C A002415 a(n) = number of Dyck (n+1)-paths with exactly n-1 peaks. - David Callan 
               (callan(AT)stat.wisc.edu), Sep 20 2007
%C A002415 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,...]. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), 
               Mar 05 2009]
%C A002415 4-dimensional square numbers. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), 
               Mar 05 2009]
%D A002415 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002415 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002415 O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 
               241.
%D A002415 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 195.
%D A002415 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, 
               Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).
%D A002415 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 A002415 Franz, Reinhard O. W. and Earnshaw, Berton A. A constructive enumeration 
               of meanders. Ann. Comb. 6 (2002), no. 1, 7-17.
%D A002415 G. Kreweras, Traitemant simultane du "Probleme de Young" et du "Probleme 
               de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle. 
               Institut de Statistique, Universit\'{e} de Paris, 10 (1967), 23-31.
%D A002415 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; 
               see p. 238.
%H A002415 T. D. Noe, <a href="b002415.txt">Table of n, a(n) for n=0..1000</a>
%H A002415 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002415 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A002415 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A002415 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A002415 H. Bottomley, <a href="a2415.gif">Illustration of initial terms</a>
%H A002415 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RiemannTensor.html">Link to a section of The World of Mathematics.</
               a>
%H A002415 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A002415 G.f.: x^2*(1+x)/(1-x)^5.
%F A002415 a(n) = sum(i = 0 to 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 - Henry Bottomley 
               (se16(AT)btinternet.com), Oct 19 2000
%F A002415 a(n)=2*C(n+2, 4)-C(n+1, 3). - Paul Barry (pbarry(AT)wit.ie), Mar 04 2003
%F A002415 a(n)=C(n+2, 4)+C(n+3, 4). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2003
%F A002415 A002415[n-1]=C[n+3, 5]-(C[n, 5]-C[n, 4]-2*C[n, 3]-C[n, 2]). - Labos E. 
               (labos(AT)ana.sote.hu), Apr 30 2003
%F A002415 a(n)=sum(k=1, n, sum(i=1, k-1, i^2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 15 2003
%F A002415 Convolution of natural numbers (A001477) with squares (A000290) - Graeme 
               McRae (g_m(AT)mcraefamily.com), Jun 06 2006
%F A002415 a(n) = n C(n+1, 3)/2 = C(n+1, 3)C(n+1,2)/(n+1) - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), 
               Jul 06 2006
%F A002415 a(n) = A006011(n)/3 = A008911(n)/2 = A047928(n-1)/12 = A083374(n-1)/6. 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007
%F A002415 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 (pbala(AT)toucansurf.com), Sep 21 2007
%F A002415 a(n)=C(n^2,2)/6,n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 07 2008
%F A002415 Starting n (-2,-1,0,1,2,...) a(n)=C(n+3,3)+2*C(n+3,4) [From Borislav 
               St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
%p A002415 a:=n->sum(sum(n^2/12, j=2..n),k=0..n): seq(a(n), n=0..38); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007
%p A002415 A002415:=-(1+z)/(z-1)**5; [S. Plouffe in his 1992 dissertation. Gives 
               sequence without initial zeros.]
%p A002415 seq(binomial(n^2,2)/6,n=0..38); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 07 2008
%p A002415 a:=n->(sum((numbperm(n,3)), j=2..n)):seq(a(n)/12, n=1..39); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 12 2008
%p A002415 a:=n->sum((n-j)^2*j,j=0..n): seq(a(n), n=0..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 07 2008
%p A002415 a:=n->(sum((numbcomp(n,4)), j=3..n))/2:seq(a(n), n=2..40); [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
%t A002415 Table[(n^4 -n^2 )/12, {n,0, 40}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 21 2007
%t A002415 s1=s2=s3=0;lst={};Do[a=n+(n+1);s1+=a;s2+=s1;s3+=s2;AppendTo[lst,s3],{n,
               0,5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 
               04 2009]
%o A002415 (PARI) a(n)=n^2*(n^2-1)/12
%o A002415 (PARI) a(n)=sum(k=1,n,sum(m=1,k,sum(i=1,m,(2*i-1)))) - Alexander R. Povolotsky 
               (pevnev(AT)juno.com), Nov 05 2007
%o A002415 sage: [lucas_number1(3,n^2,n^2)/12 for n in xrange(0,39)] - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008
%Y A002415 a(n)= ((-1)^n)*A053120(2*n, 4)/8 (one eighth of fifth unsigned column 
               of Chebyshev T-triangle, zeros omitted). Cf. A001296.
%Y A002415 Second row of array A103905.
%Y A002415 Third column of Narayana numbers A001236.
%Y A002415 Cf. A006011, A008911, A047928, A083374.
%Y A002415 Cf. A006542, A047819, A107891.
%Y A002415 Partial sums of A000330. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), 
               Mar 05 2009]
%Y A002415 Sequence in context: A055455 A050768 A063488 this_sequence A052515 A067117 
               A119365
%Y A002415 Adjacent sequences: A002412 A002413 A002414 this_sequence A002416 A002417 
               A002418
%K A002415 nonn,easy,nice
%O A002415 0,4
%A A002415 N. J. A. Sloane (njas(AT)research.att.com).
%E A002415 More terms from Larry Reeves (larryr(AT)acm.org), Oct 19 2000
%E A002415 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009