0,4

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 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

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

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

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

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

a(n) = number of squares with corners on an n X n grid. See also A024206, A108279.

Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 12 2005

Number of distinct components of the Riemann curvature tensor. - Gene Ward Smith (genewardsmith(AT)gmail.com), 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 DELEHAM (kolotoko(AT)wanadoo.fr), 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 R. Janjic (agnus(AT)blic.net), Sep 19 2007

a(n) = number of Dyck (n+1)-paths with exactly n-1 peaks. - David Callan (callan(AT)stat.wisc.edu), 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,...]. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]

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

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

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

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, Kekule 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

Franz, Reinhard O. W. and Earnshaw, Berton A. A constructive enumeration of meanders. Ann. Comb. 6 (2002), no. 1, 7-17.

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.

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

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for sequences related to linear recurrences with constant coefficients

Milan Janjic, Two Enumerative Functions

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

H. Bottomley, Illustration of initial terms

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to Chebyshev polynomials.

G.f.: x^2*(1+x)/(1-x)^5.

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

a(n)=2*C(n+2, 4)-C(n+1, 3). - Paul Barry (pbarry(AT)wit.ie), Mar 04 2003

a(n)=C(n+2, 4)+C(n+3, 4). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2003

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

a(n)=sum(k=1, n, sum(i=1, k-1, i^2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 15 2003

Convolution of natural numbers (A001477) with squares (A000290) - Graeme McRae (g_m(AT)mcraefamily.com), Jun 06 2006

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

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

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

a(n)=C(n^2,2)/6,n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008

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]

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

A002415:=-(1+z)/(z-1)**5; [S. Plouffe in his 1992 dissertation. Gives sequence without initial zeros.]

seq(binomial(n^2, 2)/6, n=0..38); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008

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

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

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]

Table[(n^4 -n^2 )/12, {n, 0, 40}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007

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]

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

(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

sage: [lucas_number1(3, n^2, n^2)/12 for n in xrange(0, 39)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008

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 A001236.

Cf. A006011, A008911, A047928, A083374.

Cf. A006542, A047819, A107891.

Partial sums of A000330. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]

Adjacent sequences: A002412 A002413 A002414 this_sequence A002416 A002417 A002418

Sequence in context: A055455 A050768 A063488 this_sequence A052515 A067117 A119365

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Larry Reeves (larryr(AT)acm.org), Oct 19 2000

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009