A006011 -id:A006011 - cp's OEIS frontend

A008911 a(n) = n^2*(n^2 - 1)/6.

0, 0, 2, 12, 40, 100, 210, 392, 672, 1080, 1650, 2420, 3432, 4732, 6370, 8400, 10880, 13872, 17442, 21660, 26600, 32340, 38962, 46552, 55200, 65000, 76050, 88452, 102312, 117740, 134850, 153760, 174592, 197472, 222530, 249900, 279720, 312132

Offset: 0

N. J. A. Sloane, R. K. Guy

Number of equilateral triangles in rhombic portion of side n+1 in hexagonal lattice.
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
Sum of squared distances on n X n board between n queens each on its own row and column. - Zak Seidov, Sep 04 2002
For queens "each on its column and row" the sum of squared distances does not depend on configuration - while sum of distances does.
Number of cycles of length 3 in the bishop's graph associated with an (n+1) X (n+1) chessboard. - Anton Voropaev (anton.n.voropaev(AT)gmail.com), Feb 01 2009
a(n) is number of ways to place 3 queens on an (n+1) X (n+1) chessboard so that they diagonally attack each other exactly 3 times. The maximal possible attack number, p=binomial(k,2)=3 for k=3 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - Antal Pinter, Dec 27 2015
From a(1), convolution of the oblong numbers (A002378) with the odd numbers (A005408). - Bruno Berselli, Oct 24 2016
Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, p and |q-p|. - Wesley Ivan Hurt, Apr 15 2018

			a(2)=2 because on 2 X 2 board queens "each on its column and row" may take only two angular cells, then squared distance is 1^2+1^2=2. a(3)=12 because on 3 X 3 board queens "each on its column and row" make only two essentially distinct configurations: {1,2,3}, {1,3,2} and in both cases the sum of three squared distances is 12.
G.f.: 2*x^2 + 12*x^3 + 40*x^4 + 100*x^5 + 210*x^6 + 392*x^7 + 672*x^8 + ...

James 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 Problem 6).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics, 1999.
James Propp, Updated article, 2009.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002415, A006011, A047928, A083374.
Cf. A002378, A005408.
Convolution of the oblong numbers with the even numbers: A033488.

List([0..40], n-> n^2*(n^2-1)/6); # G. C. Greubel, Sep 13 2019

[n^2*(n^2-1)/6: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011

A008911 := n->n^2*(n^2-1)/6; seq(A008911(n), n=0..40);

a[m_]:= m^2(m^2-1)/6;
Binomial[Range[0,40]^2, 2]/3 (* G. C. Greubel, Sep 13 2019 *)

PARI
```
{a(n) = n^2*(n^2-1)/6};
```

[n^2*(n^2-1)/6 for n in (0..40)] # G. C. Greubel, Sep 13 2019

G.f.: 2*x^2*(1+x)/(1-x)^5.
a(n) = 2*A002415(n) = A047928(n-1)/6 = A083374(n-1)/3 = A006011(n)*2/3. - Zerinvary Lajos, May 09 2007
a(n) = n*binomial(n+1,3). - Martin Renner, Apr 03 2011
a(n+1) = (n+1)*A000292(n). - Tom Copeland, Sep 13 2011
From G. C. Greubel, Sep 13 2019: (Start)
a(n) = binomial(n^2,2)/3.
E.g.f.: x^2*(6 + 6*x + x^2)*exp(x)/6. (End)
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=2} 1/a(n) = 21/2 - Pi^2.
Sum_{n>=2} (-1)^n/a(n) = (Pi^2 - 9)/2. (End)
a(n) = Sum_{j=0..n-1} binomial(n,2) + binomial(n,3). - Detlef Meya, Jan 20 2024

A089574 Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).

Original entry on oeis.org

5, 32, 113, 299, 664, 1309, 2366, 4002, 6423, 9878, 14663, 21125, 29666, 40747, 54892, 72692, 94809, 121980, 155021, 194831, 242396, 298793, 365194, 442870, 533195, 637650, 757827, 895433, 1052294, 1230359, 1431704, 1658536, 1913197

Offset: 1

Alford Arnold, Dec 29 2003; extended May 04 2005

The diagonals are finite and sum to A047970.
Values appear to be a transformation of A006468 (rooted planar maps). Also known as well-labeled trees (cf. A000168).
First differences of the conjectured polynomial formula for A006468. [From R. J. Mathar, Jun 26 2010]

			The array begins
1
2
4
7 1
11 5
16 14 2
22 30 12
29 55 39 5
37 91 95 32 1

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A105552, A006468.
Cf. A006011, A034261.
Cf. A000124 (column 1), A000330 (column 2), A086602 (column 3), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13).

Row sums are powers of 2.
a(n) = A000330(n) + A006011(n+1) + A034263(n-1).
a(n)= +6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6). G.f.: x*(5+2*x-4*x^2+x^3)/(x-1)^6. a(n) = n*(n+1)*(4*n^3+51*n^2+159*n+86)/120. [From R. J. Mathar, Jun 26 2010]

Extended beyond a(8) by R. J. Mathar, Jun 26 2010

A107600 Column 5 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

1, 18, 101, 357, 978, 2274, 4711, 8954, 15915, 26806, 43197, 67079, 100932, 147798, 211359, 296020, 406997, 550410, 733381, 964137, 1252118, 1608090, 2044263, 2574414, 3214015, 3980366, 4892733, 5972491, 7243272, 8731118, 10464639

Offset: 9

Alford Arnold, May 17 2005

The sequences in A089574 count ordered partitions. Sequence A001296 can be associated with 9 = 3+3+3. Six times sequence A005585, associated with 10 = 3+3+2+2. The other three sequences comprising A107600 are generated in A034261 and can be associated with 10 = 5 + 5 = 4 + 4 + 2 = 2 + 2 + 2 + 2 + 2.

			A107600(n) can be constructed from five other sequences as follows:
1...7...25...65...140.......A001296
....1...11...56...196.......A034264
....6...42..162...462.......6.*.A005585.
....3...18...60...150.......A006011
....1....5...14....30.......A000330
therefore
1..18..101..357...978.......A107600

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A034261, A089574, A000330, A001296, A005585, A006011, A034264.

a:= n-> `if` (n<9, 0, (92292 +(-6580 +(-5745 +(1535 +(-147+5*n) *n) *n) *n) *n) *n /720 -218): seq(a(n), n=9..45); # Alois P. Heinz, Nov 06 2009

Select[CoefficientList[Series[(x^5-5x^4+7x^3+4x^2-11x-1)x^9/(x-1)^7, {x,0,50}],x],#>0&] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1,18,101,357,978,2274,4711},42] (* Harvey P. Dale, May 01 2011 *)

G.f.: (x^5 -5*x^4 +7*x^3 +4*x^2 -11*x -1) *x^9 /(x-1)^7. - Alois P. Heinz, Nov 06 2009

More terms from Alois P. Heinz, Nov 06 2009

A124428 Triangle, read by rows: T(n,k) = binomial(floor(n/2),k)*binomial(floor((n+1)/2),k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 6, 3, 1, 9, 9, 1, 1, 12, 18, 4, 1, 16, 36, 16, 1, 1, 20, 60, 40, 5, 1, 25, 100, 100, 25, 1, 1, 30, 150, 200, 75, 6, 1, 36, 225, 400, 225, 36, 1, 1, 42, 315, 700, 525, 126, 7, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 56, 588, 1960, 2450, 1176, 196, 8

Offset: 0

Paul D. Hanna, Oct 31 2006

Row sums form A001405, the central binomial coefficients: C(n,floor(n/2)). The eigenvector of this triangle is A124430.
T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) having k peaks. Example: T(5,2)=3 because, denoting U=(1,1), D=(1,-1), H=(1,0), we have HUDUD, UDHUD, and UDUDH. - Emeric Deutsch, Jun 01 2011
From Emeric Deutsch, Jan 18 2013: (Start)
T(n,k) is the number of Dyck prefixes of length n having k peaks. Example: T(5,2)=3 because we have (UD)(UD)U, (UD)U(UD), and U(UD)(UD); the peaks are shown between parentheses.
T(n,k) is the number of Dyck prefixes of length n having k ascents and descents of length >= 2. Example: T(5,2)=3 because we have (UU)(DD)U, (UU)D(UU), and (UUU)(DD); the ascents and descents of length >= 2 are shown between parentheses. (End)
T(n,k) is the number of noncrossing partitions of [n] having n-k blocks, such that the nontrivial blocks are of type {a,b}, with a < = n/2 and b > n/2. Such partitions  have k nontrivial blocks, uniquely determined by the choice of k first elements among floor(n/2) elements, and the choice of k second elements among floor((n+1)/2) elements. Indeed, by planarity, any two blocs {a,b} and {c,d} satisfy a < c iff b > d. - Francesca Aicardi Nov 03 2022

			Triangle begins:
  1;
  1;
  1,   1;
  1,   2;
  1,   4,   1;
  1,   6,   3;
  1,   9,   9,   1;
  1,  12,  18,   4;
  1,  16,  36,  16,   1;
  1,  20,  60,  40,   5;
  1,  25, 100, 100,  25,   1;
  1,  30, 150, 200,  75,   6;
  1,  36, 225, 400, 225,  36,   1; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See paragraph 2.1.

Cf. A001405 (row sums), A056953, A026003, A124429 (antidiagonal sums), A124430 (eigenvector), A191521.

Columns = A002378, A006011, A006542, etc.

[[Binomial(Floor(n/2), k)*Binomial(Floor((n+1)/2),k): k in [0..Floor(n/2)]]: n in [0..15]]; // G. C. Greubel, Feb 24 2019

Table[Binomial[Floor[n/2], k]*Binomial[Floor[(n+1)/2], k], {n, 0, 15}, {k, 0, Floor[n/2]}]//Flatten (* G. C. Greubel, Feb 24 2019 *)

T(n,k)=binomial(n\2,k)*binomial((n+1)\2,k)

[[binomial(floor(n/2),k)*binomial(floor((n+1)/2),k) for k in (0..floor(n/2))] for n in (0..15)] # G. C. Greubel, Feb 24 2019

A056953(n) = Sum_{k=0..floor(n/2)} k!*T(n,k).

A026003(n) = Sum_{k=0..floor(n/2)} 2^k*T(n,k).

A108538 Column 10 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

1, 64, 731, 4553, 20155, 71272, 214653, 572743, 1389702, 3122752, 6585183, 13162741, 25131718, 46115029, 81722067, 140429357, 234772177, 382932581, 610826859, 954815625, 1465182669, 2210554686, 3283463257, 4807283267, 6944818576, 9908846494, 13974977743, 19497238421, 26926835328

Offset: 0

Alford Arnold, Jul 05 2005

A109820 can be decomposed into 30 sequences. These 30 associated sequences can be inferred from the 30 ways of partitioning the number nine: 9 81 72 63 54 ... the complete listing is available in the Handbook of Mathematical Functions (1964) p. 831. Consider, for example, the three ways of partitioning the number three: 3, 21 and 111; prepend each partition then add one to each value - yielding 44, 332 and 2222. These "associated" partitions are then used to derive the associated sequences. 44 => A000330, 332 => A006011 and 2222 => A034263. Summing these three sequences yields A089574.

			a(1) = 1 because the only associated partition 4444 for n = 16 cannot be permuted.
a(2) = 64 because the associated partitions can be permuted in 3 + 4 + 12 + 9 + 20 + 10 + 6 ways when n = 17.

Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000330 (column 2), A086602 (column 3), A089574 (column 4), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13)

G.f. 1+64*x -x^2*(-731 +4219*x -13765*x^2 +30910*x^3 -49804*x^4 +58458*x^5 -50237*x^6 +31394*x^7 -13931*x^8 +4171*x^9 -757*x^10 +63*x^11)/(x-1)^12 . - R. J. Mathar, Aug 28 2018

Extended by R. J. Mathar, Aug 28 2018

A112851 a(n) = (n-1)n(n+1)(n+2)(2*n+1)/40.

Original entry on oeis.org

0, 0, 3, 21, 81, 231, 546, 1134, 2142, 3762, 6237, 9867, 15015, 22113, 31668, 44268, 60588, 81396, 107559, 140049, 179949, 228459, 286902, 356730, 439530, 537030, 651105, 783783, 937251, 1113861, 1316136, 1546776, 1808664, 2104872, 2438667, 2813517, 3233097

Offset: 0

Alford Arnold, Sep 24 2005

A112851 is the fourth sequence in A112852.

Also the Wiener index of the (n-1)-triangular grid graph (indexed so the 0-triangular grid graph is the singleton). - Eric W. Weisstein, Sep 08 2017

L. B. W. Jolley, Summation of Series, Dover. N.Y., 1961, eq. (54), page 10.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Partial sums of sequence A006011.

Cf. A000217, A033428, A112851, A112852.

[(n-1)*n*(n+1)*(n+2)*(2*n+1)/40: n in [0..40]]; // Vincenzo Librandi, Feb 06 2014

a:=n->sum(j^4-j^2, j=0..n)/4: seq(a(n), n=0..36); # Zerinvary Lajos, May 08 2008

Table[(n - 1) n (n + 1)(n + 2)(2 n + 1)/40, {n, 0, 30}] (* Josh Locker *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 3, 21, 81, 231}, 40] (* Harvey P. Dale, Oct 28 2014 *)

for(n=0,50, print1((n-1)*n*(n+1)*(n+2)*(2*n+1)/40, ", ")) \\ G. C. Greubel, Jul 23 2017

concat(vector(2), Vec(3*x^2*(1 + x) / (1 - x)^6 + O(x^30))) \\ Colin Barker, Sep 08 2017

4*a(n+1) = 1*2^2*3 + 2*3^2*4 + 3*4^2*5 + ... (n terms). [Jolley]
a(n) = Sum_{i=0..n} A000217(i-1)*A000217(i), where A000217(-1)=0. - Bruno Berselli, Feb 05 2014
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. - Harvey P. Dale, Oct 28 2014
G.f.: 3*x^2*(1 + x) / (1 - x)^6. - Colin Barker, Sep 08 2017
a(n) = (1/2) * Sum_{k=0..n} C(k^2,2). - Wesley Ivan Hurt, Sep 23 2017
a(n) = Sum_{i=0..n} A000217(i)*A033428(n-i). - Bruno Berselli, Mar 06 2018
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=2} 1/a(n) = 40*(16*log(2) - 11)/9.
Sum_{n>=2} (-1)^n/a(n) = 20*(8*Pi - 25)/9. (End)

More terms from Josh Locker (jlocker(AT)mail.rochester.edu) and Michael W. Motily (mwm5036(AT)psu.edu), Oct 04 2005

A163933 Third right hand column of triangle A163932.

Original entry on oeis.org

11, 105, 510, 1750, 4830, 11466, 24360, 47520, 86625, 149435, 246246, 390390, 598780, 892500, 1297440, 1844976, 2572695, 3525165, 4754750, 6322470, 8298906, 10765150, 13813800, 17550000, 22092525, 27574911, 34146630, 41974310

Offset: 3

Johannes W. Meijer, Aug 13 2009

Harvey P. Dale, Table of n, a(n) for n = 3..999
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A048994 (Stirling1).
Equals the third right hand column of triangle A163932.
A000217 and A006011 are the first and second right hand columns.

nmax:=30; with(combinat, stirling1): for n from 1 to nmax do for m from 1 to n do a(n,m):=(-1)^(n+m)*(m)*(m+1)*stirling1(n+1,m+1)/2 od: od: seq(a(n,n-2),n=3..nmax);

Table[(n-2)(n-1)StirlingS1[n+1,n-1]/2,{n,3,30}] (* Harvey P. Dale, Oct 09 2011 *)

a(n) = (n-2)*(n-1)*Stirling1(n+1,n-1)/2.
G.f.: z^3*(11 + 6*z^2 + 28*z)/(1-z)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 9. - Chai Wah Wu, Jan 25 2021

A218979 Numbers n such that some sum of n consecutive positive cubes is a square.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 25, 27, 28, 29, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 57, 59, 60, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 76, 77, 79, 81, 82, 83, 85, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99

Offset: 1

Michel Marcus, Nov 08 2012

nonn

The trivial solutions with x = 0 and x = 1 are not considered here.
Numbers n such that x^3 + (x+1)^3 + ... + (x+n-1)^3 = y^2 has nontrivial solutions over the integers.
The nontrivial solutions are found by solving Y^2 = X^3 + d(n)*X with d(n) = n^2*(n^2-1)/4 (A006011), Y = n*y and X = n*x + (1/2)*n*(n-1). [Corrected by Derek Orr, Aug 30 2014]
x^3 + (x+1)^3 + ... + (x+n-1)^3 = y^2 can also be written as y^2 = n(x + (n-1)/2)(n(x + (n-1)/2) + x(x-1)). - Vladimir Pletser, Aug 30 2014
There are 892 triples (n,x,y), with n and x less than 10^5 (1 < n,x < 10^5), which are nontrivial solutions of x^3 + (x+1)^3 + ... + (x+n-1)^3 = y^2 (note that (n,x,y) corresponds to (M,a,c) in A253679, A253680, A253681, A253707, A253708, A253709, A253724, A253725). - Vladimir Pletser, Jan 10 2015

			See "Examples of triples" link.

Michel Marcus, Examples of triples up to n=50
Vladimir Pletser, Triplets (n, x, y) with n,x less than 10^5
Vladimir Pletser, Number of terms, first term and square root of sums of consecutive cubed integers equal to integer squares, Research Gate, 2015.
Vladimir Pletser, Fundamental solutions of the Pell equation X^2-(sigma^4-delta^4)Y^2=delta^4 for the first 45 solutions of the sums of consecutive cubed integers equalling integer squares, Research Gate, 2015. See Reference 19.
V. Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015.
R. J. Stroeker, On the sum of consecutive cubes being a perfect square, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307.

Cf. A116108, A116145, A126200, A126203, A163392, A163393, A253679, A253680, A253681, A253707, A253708, A253709.

a(n)=for(x=2,10^7, /* note this limit only generates the terms in the data section */ X = n*x + (1/2)*n*(n-1); d=n^2*(n^2-1)/4;if(issquare(X^3+d*X),return(x)))
n=1;while(n<100,if(a(n),print1(n,", "));n++) \\ Derek Orr, Aug 30 2014

Name changed, a(1) = 1 prepended and a(39)-a(68) from Derek Orr, Aug 30 2014

More terms for 50Vladimir Pletser, Jan 10 2015

A228317 The hyper-Wiener index of the triangular graph T(n) (n >= 1).

Original entry on oeis.org

0, 0, 3, 21, 75, 195, 420, 798, 1386, 2250, 3465, 5115, 7293, 10101, 13650, 18060, 23460, 29988, 37791, 47025, 57855, 70455, 85008, 101706, 120750, 142350, 166725, 194103, 224721, 258825, 296670, 338520, 384648, 435336, 490875, 551565, 617715, 689643

Offset: 1

Emeric Deutsch, Aug 26 2013

The triangular graph T(n) is the graph whose vertices represent the 2-subsets of {1,2,...,n} and two vertices are adjacent provided the corresponding 2-subsets have a nonempty intersection.

The triangular graph T(n) is a strongly regular graph with parameters n*(n-1)/2, 2*(n-2), n-2, and 4 (see the Brualdi and Ryser reference, Theorem 5.2.4).

R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

G. G. Cash, Relationship between the Hosoya polynomial and the hyper-Wiener index, Applied Mathematics Letters, 15(7) (2002), 893-895.
Eric Weisstein's World of Mathematics, Triangular Graph.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001296, A006011.

a := proc (n) options operator, arrow: (1/8)*n*(n-1)*(n-2)*(3*n-5) end proc: seq(a(n), n = 1 .. 38);

LinearRecurrence[{5,-10,10,-5,1},{0,0,3,21,75},40] (* Harvey P. Dale, Feb 23 2023 *)

a(n) = n*(n - 1)*(n - 2)*(3*n - 5)/8.
G.f.: 3*x^3*(1 + 2*x)/(1 - x)^5.
The Hosoya-Wiener polynomial of T(n) is (1/8)*n*(n - 1)*(4 + 4*(n-2)*t + (n - 2)*(n - 3)*t^2).
a(n) = 3*A001296(n-2) for n >= 2. - R. J. Mathar, Mar 05 2017

A131635 Triangle T(n,m)=mnbinomial(m+n,m)^2/(2*(m+n)) read by rows.

Original entry on oeis.org

1, 3, 18, 6, 60, 300, 10, 150, 1050, 4900, 15, 315, 2940, 17640, 79380, 21, 588, 7056, 52920, 291060, 1280664, 28, 1008, 15120, 138600, 914760, 4756752, 20612592, 36, 1620, 29700, 326700, 2548260, 15459444, 77297220, 331273800, 45, 2475, 54450

Offset: 1

R. J. Mathar, Sep 05 2007

First two columns are essentially A000217 and A006011.

			Triangle is symmetric in the two indices and starts
1,
3, 18,
6, 60, 300,
10, 150, 1050, 4900,
15, 315, 2940, 17640, 79380,
21, 588, 7056, 52920, 291060, 1280664,

V. J. W. Guo and J. Zeng, A note on two identities arising from enumeration of convex polyominoes, J. Comp. Appl. Math. 180 (2005) pp 413-423.

a := proc(n,m) m*n*(binomial(m+n,n))^2/2/(m+n) ; end: for n from 1 to 10 do for m from 1 to n do printf("%d, ",a(n,m)) ; od: od:

Flatten[Table[m*n*Binomial[m+n,m]^2/(2(m+n)),{n,10},{m,n}]] (* Harvey P. Dale, Dec 24 2011 *)

A131635(n,m) = m*n*binomial(m+n,m)^2/(2*(m+n))

T(n,m)=m*n*A000290(A007318(n+m,m))/[2(m+n)].

cp's OEIS Frontend

A008911 a(n) = n^2*(n^2 - 1)/6.

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A089574 Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).

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A107600 Column 5 of array illustrated in A089574 and related to A034261.

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A124428 Triangle, read by rows: T(n,k) = binomial(floor(n/2),k)*binomial(floor((n+1)/2),k).

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A108538 Column 10 of array illustrated in A089574 and related to A034261.

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A112851 a(n) = (n-1)*n*(n+1)*(n+2)*(2*n+1)/40.

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A112851 a(n) = (n-1)n(n+1)(n+2)(2*n+1)/40.

A131635 Triangle T(n,m)=mnbinomial(m+n,m)^2/(2*(m+n)) read by rows.