%I A000537 M4619 N1972
%S A000537 0,1,9,36,100,225,441,784,1296,2025,3025,4356,6084,8281,11025,
%T A000537 14400,18496,23409,29241,36100,44100,53361,64009,76176,90000,105625,
%U A000537 123201,142884,164836,189225,216225,246016,278784,314721,354025
%N A000537 Sum of first n cubes; or n-th triangular number squared.
%C A000537 Number of parallelograms in an n X n rhombus - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), 
               May 14 2000.
%C A000537 Or, number of orthogonal rectangles in an n X n checkerboard, or rectangles 
               in an n X n array of squares. - Jud McCranie, Feb 28 2003. Compare 
               A085582.
%C A000537 Also number of 2-dimensional cage assemblies (cf. A059827, A059860).
%C A000537 The n-th triangular number T(n)=sum_r(1, n)=n(n+1)/2 satisfies the relations: 
               (i) T(n) + T(n-1)=n^2 and (ii) T(n) - T(n-1)=n from definition, so 
               that n^2*n=n^3={T(n)}^2 - {T(n-1)}^2 and thus summing telescopingly 
               over n we have sum_{ r = 1..n } r^3 = {T(n)}^2 = (1+2+3+...+n)^2 
               = (n*(n+1)/2)^2. - Lekraj Beedassy (blekraj(AT)yahoo.com), May 14 
               2004
%C A000537 Number of 4-tuples of integers from {0,1,...,n}, without repetition, 
               whose last component is strictly bigger than the others. Number of 
               4-tuples of integers from {1,...,n}, with repetition, whose last 
               component is greater than or equal to the others.
%C A000537 Number of ordered pairs of two element subsets of {0,1,...,n} without 
               repetition. Number of ordered pairs of 2-element multisubsets of 
               {1,...,n} with repetition.
%C A000537 1^3 + 2^3 + 3^3 +...+ n^3=(1+2+3+...+n)^2
%C A000537 a(n) is the number of parameters needed in general to know the Riemannian 
               metric g of an n-dimensional Riemannian manifold (M,g), by knowing 
               all its second derivatives; even though to know the curvature tensor 
               R requires (due to symmetries) (n^2)*(n^2-1)/12 parameters, a smaller 
               number (and a 4-dimensional pyramidal number). - Jonathan Vos Post 
               (jvospost3(AT)gmail.com), May 05 2006
%C A000537 Also number of hexagons with vertices in an hexagonal grid with n points 
               in each side. - Ignacio Larrosa Canestro (ilarrosa(AT)mundo-r.com), 
               Oct 15 2006
%C A000537 Number of permutations of n distinct letters (ABCD...) each of which 
               appears twice with 4 and n-4 fixed points. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 09 2006
%C A000537 With offset 1 = binomial transform of [1, 8, 19, 18, 6,...]. [From Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Dec 03 2008]
%C A000537 Sum(k>0,1/a(k))=(4/3)*(Pi^2-9) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Sep 20 2009]
%D A000537 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000537 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000537 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A000537 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 813.
%D A000537 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, p. 110ff.
%D A000537 Marcel Berger, Encounter with a Geometer, Part II, Notices of the American 
               Mathematical Society, Vol. 47, No. 3, (March 2000), pp. 326-340. 
               [About the work of Mikhael Gromov].
%D A000537 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
%D A000537 John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, 
               pp. 36, 58.
%D A000537 Clifford Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind 
               and Meaning," Oxford University Press, 2001, p. 325.
%D A000537 D. Wells, You Are A Mathematician, "Counting rectangles in a rectangle", 
               Problem 8H, pp. 240; 254, Penguin Books 1995.
%H A000537 T. D. Noe, <a href="b000537.txt">Table of n, a(n) for n=0..1000</a>
%H A000537 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000537 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A000537 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 A000537 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 A000537 M. Azaola and F. Santos, <a href="http://personales.unican.es/santosf/
               Articulos/">The number of triangulations of the cyclic polytope C(n,
               n-4)</a>, Discrete Comput. Geom., 27 (2002), 29-48 (see Prop. 4.2(b)).
%H A000537 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind 
               and Meaning," <a href="http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an\
               =0983.00008&format=complete">Zentralblatt review</a>
%H A000537 G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">
               Operate on "n^3"</a>
%F A000537 a(n) = (n*(n+1)/2)^2, that is, 1^3 + 2^3 + 3^3 +...+ n^3 = (1+2+3+...+n)^2. 
               G.f.: (x+4*x^2+x^3)/(1-x)^5.
%F A000537 a(n) = Sum [ Sum ( 1 + Sum (6*n) ) ]. - Xavier Acloque, Jan 21 2003
%F A000537 Sum(j=1, n, j*triangle(n)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 
               28 2003
%F A000537 a(n) = Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}] - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Oct 24 2004
%F A000537 Equals A000217(n)^2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 26 2007
%F A000537 a(n)=A035287(n)/4 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 
               09 2007
%F A000537 This sequence could be obtained from the general formula n*(n+1)*(n+2)*(n+3)* 
               ...* (n+k) *(n*(n+k) + (k-1)*k/6)/((k+3)!/6) at k=1. - Alexander 
               R. Povolotsky (pevnev(AT)juno.com), May 17 2008
%F A000537 G.f.: x*F(3,3;1;x); [From Paul Barry (pbarry(AT)wit.ie), Sep 18 2008]
%p A000537 (n*(n+1)/2)^2;
%p A000537 [seq((binomial(n,2))^2,n=1..35)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 24 2006
%p A000537 [seq (stirling2(n+1,n)*binomial(n+1,2),n=0..34)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 06 2006
%p A000537 a:=array(0...34): a[0]:=0: a[1]:=1:print(0,a[0]); print(1,a[1]); for 
               i from 2 to 34 do a[i]:= a[i-1]+(i^3):print(i,a[i]); od: - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 26 2007
%p A000537 a:=n->sum(sum(n^2/4, j=0..n),k=0..n): seq(a(n), n=0..34); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007
%p A000537 seq(sum(sum(lcm(k,j),j=1..n), k=0..n), n=0..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 01 2007
%p A000537 A000537:=-(1+4*z+z**2)/(z-1)**5; [S. Plouffe in his 1992 dissertation 
               for sequence without initial zero.]
%p A000537 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^3 od: seq(a[n], 
               n=0..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 
               2008
%p A000537 a:=n->sum(k*sum(k, k=0..n), k=0..n):seq(a(n), n=0...34); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Aug 01 2008
%p A000537 a:=n->sum(k*sum(k, k=0..n), k=0..n):seq(a(n), n=0...34); [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Aug 09 2008]
%t A000537 Table[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}], {n, 0, 10}]
%t A000537 Table[StirlingS2[i+1, i]*(-StirlingS1[i+1, i]), {i,0, 34}] - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2007
%t A000537 s = 0; lst = {s}; Do[s += n^3; AppendTo[lst, s], {n, 1, 42, 1}]; lst 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
%o A000537 (PARI) a(n)=(n*(n+1)/2)^2
%o A000537 (PARI) t(n)=n*(n+1)/2 for(i=1,30,print1(","sum(j=1,i,j*t(i))))
%o A000537 (PARI) a(n)=sum(m=1,n,sum(i=m*(m+1)/2-m+1,m*(m+1)/2,(2*i-1))) - Alexander 
               R. Povolotsky (pevnev(AT)juno.com), Nov 05 2007
%Y A000537 Convolution of A000217 and A008458. Cf. A000330, A006003, A000538.
%Y A000537 Row sums of triangles A094414 and A094415.
%Y A000537 Second column of triangle A008459.
%Y A000537 Row 3 of array A103438.
%Y A000537 Cf. A000217, A002415.
%Y A000537 Cf. A101102, A101097, A101094, A024166, A000578.
%Y A000537 Sequence in context: A134537 A066647 A085037 this_sequence A114286 A098928 
               A139469
%Y A000537 Adjacent sequences: A000534 A000535 A000536 this_sequence A000538 A000539 
               A000540
%K A000537 nonn,easy,nice
%O A000537 0,3
%A A000537 N. J. A. Sloane (njas(AT)research.att.com).
%E A000537 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009