A077414 -id:A077414 - cp's OEIS frontend

A077415 a(n) = n(n+2)(n-2)/3.

0, 5, 16, 35, 64, 105, 160, 231, 320, 429, 560, 715, 896, 1105, 1344, 1615, 1920, 2261, 2640, 3059, 3520, 4025, 4576, 5175, 5824, 6525, 7280, 8091, 8960, 9889, 10880, 11935, 13056, 14245, 15504, 16835, 18240, 19721, 21280, 22919, 24640, 26445

Offset: 2

Wolfdieter Lang, Nov 29 2002

a(n) is the number of independent components of a 3-tensor t(a,b,c) which satisfies t(a,b,c)=t(b,a,c) and sum(t(a,a,c),a=1..n)=0 for all c and t(a,b,c)+t(b,c,a)+t(c,a,b)=0, with a,b,c range 1..n. (3-tensor in n-dimensional space which is symmetric and traceless in one pair of its indices and satisfies the cyclic identity.)
Number of standard tableaux of shape (n-1,2,1) (n>=3). - Emeric Deutsch, May 13 2004
Zero followed by partial sums of A028387, starting at n=1. - Klaus Brockhaus, Oct 21 2008
For n>=4, a(n-1) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0101 (the first n-4 zeros), or, the same, a(n-1) is up-down coefficient {n,5} (see comment in A060351). - Vladimir Shevelev, Feb 14 2014
For n>=3, a(n) equals the second immanant of the (n-1) X (n-1) tridiagonal matrix with 2's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jan 08 2016

G. C. Greubel, Table of n, a(n) for n = 2..10000
Mark Roger Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A028387 (first differences), A033275 (partial sums), A060351, A077414, A084990.

[n*(n+2)*(n-2)/3: n in [2..50]]; /* or */ I:=[0,5,16,35]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jan 09 2016

seq((n^3-4*n)/3, n=2..35); # Zerinvary Lajos, Jan 20 2007

Print[Table[Sum[(-1)^i*2^(n-2*i-1)*Binomial[n-i-1, i]*(n-2*i-2), {i, 0, Floor[(n-1)/2]}], {n, 2, 100}]] ;  (* John M. Campbell, Jan 08 2016 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 5, 16, 35}, 50] (* Vincenzo Librandi, Jan 09 2016 *)
Table[n*(n + 2)*(n - 2)/3, {n, 2, 50}] (* G. C. Greubel, Jan 18 2018 *)

{a=0; print1(a,","); for(n=1, 42, print1(a=a+n+(n+1)^2, ","))} \\ Klaus Brockhaus, Oct 21 2008

concat(0, Vec(x^3*(5-4*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 08 2015

a(n) = n*(n+2)*(n-2)/3 = A077414(n) - binomial(n+2,3) = A077414(n) - A000292(n-1).
G.f.: x^3*(5 - 4*x + x^2)/(1-x)^4.
a(n) = A084990(n-1) - 1. - Reinhard Zumkeller, Aug 20 2007
a(n) = Sum_{i=0..floor((n-1)/2)} (-1)^i * 2^(n-2*i-1) * binomial(n-i-1, i) * (n-2*i-2). - John M. Campbell, Jan 08 2016
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=3} 1/a(n) = 11/32.
Sum_{n>=3} (-1)^(n+1)/a(n) = 5/32. (End)
E.g.f.: x*(1 + exp(x)*(x^2 + 3*x - 3)/3). - Stefano Spezia, Mar 06 2024

A215862 Number of simple labeled graphs on n+2 nodes with exactly n connected components that are trees or cycles.

Original entry on oeis.org

0, 4, 19, 55, 125, 245, 434, 714, 1110, 1650, 2365, 3289, 4459, 5915, 7700, 9860, 12444, 15504, 19095, 23275, 28105, 33649, 39974, 47150, 55250, 64350, 74529, 85869, 98455, 112375, 127720, 144584, 163064, 183260, 205275, 229215, 255189, 283309, 313690, 346450

Offset: 0

Alois P. Heinz, Aug 25 2012

Partial sums of A077414. - Bruno Berselli, Jul 30 2015

			a(1) = 4:
.1-2.  .1-2.  .1-2.  .1 2.
.|/ .  .|. .  . / .  .|/ .
.3...  .3...  .3...  .3...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

A diagonal of A215861.
Regarding the sixth formula, see similar sequences listed in A241765.
Cf. A000332, A077414.

a:= n-> binomial(n+2,3)*(3*n+13)/4:
seq(a(n), n=0..40);

Table[Binomial[n+2,3] (3n+13)/4,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,4,19,55,125},40] (* Harvey P. Dale, Sep 10 2012 *)

G.f.: (x-4)*x/(x-1)^5.
a(n) = C(n+2,3)*(3*n+13)/4.
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, Sep 10 2012
a(n) = (1/n!) * Sum_{j=0..n} C(n,j)*(-1)^(n-j)*j^(n+1)*(j-1). - Vladimir Kruchinin, Jun 06 2013
a(n) = 4*A000332(n+2) - A000332(n+1). - R. J. Mathar, Aug 12 2013
a(n) = Sum_{i=0..n} (3+i)*A000217(i). - Bruno Berselli, Apr 29 2014

A267370 Partial sums of A140091.

Original entry on oeis.org

0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510

Offset: 0

Bruno Berselli, Jan 13 2016

After 0, this sequence is the third column of the array in A185874.

Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.

			The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
.  0;
.  1,  5;
.  4,  7, 10;
.  9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005581, A051744, A105163, A140091, A185874.

Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).

Magma
```
[n*(n+1)*(n+5)/2: n in [0..50]];
```

Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,6,21,48},50] (* Harvey P. Dale, Jul 18 2019 *)

PARI
```
vector(50, n, n--; n*(n+1)*(n+5)/2)
```
Sage
```
[n*(n+1)*(n+5)/2 for n in (0..50)]
```

O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
a(n) = 3*A005581(n+1).
a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)

A213761 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-5+3h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 4, 18, 15, 7, 40, 36, 24, 10, 75, 70, 54, 33, 13, 126, 120, 100, 72, 42, 16, 196, 189, 165, 130, 90, 51, 19, 288, 280, 252, 210, 160, 108, 60, 22, 405, 396, 364, 315, 255, 190, 126, 69, 25, 550, 540, 504, 448, 378, 300

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A172073.
Antidiagonal sums: A002419.
Row 1, (1,2,3,4,5,...)**(1,4,7,10,13,...): A002411.
Row 2, (1,2,3,4,5,...)**(4,7,10,13,16,...): A077414.
Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 + 6*k)/2.
Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 + 9*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....18...40....75....126
4....15...36...70....120...189
7....24...54...100...165...252
10...33...72...130...210...315
13...42...90...160...255...378

Clark Kimberling, Antidiagonals n = 1..45, flattened

Cf. A212500.

b[n_]:=n;c[n_]:=3n-2;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213761 *)
Table[t[n,n],{n,1,40}] (* A172073 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A002419 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*(3*n - 2 - (3*n - 5)*x) and g(x) = (1 - x)^4.

A223544 T(n, k) = n*k - 1.

Original entry on oeis.org

0, 1, 3, 2, 5, 8, 3, 7, 11, 15, 4, 9, 14, 19, 24, 5, 11, 17, 23, 29, 35, 6, 13, 20, 27, 34, 41, 48, 7, 15, 23, 31, 39, 47, 55, 63, 8, 17, 26, 35, 44, 53, 62, 71, 80, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 120, 11, 23, 35, 47, 59, 71, 83, 95, 107, 119, 131, 143

Offset: 1

Richard R. Forberg, Jul 19 2013

Previous name was: Triangle T(n,k), 0 < k <= n, T(n,1) = n - 1, T(n,k) = T(n,k-1) + n; read by rows.
This simple triangle arose analyzing f(x) = x/(n + e^(c/x)), for n <> 0. f(x) converges towards a rational number for large values of x, if x is rational. T(n+1,k)/(n+1)^2 equals the fractional portion of f(x) if x is large and restricted to the positive integers, c = 1 and n>=1, whereby the value of the fractional portion changes on a cycle with period n+1 (as k goes from 1 to n+1) for each n in the denominator of f(x). Other, somewhat similar triangles (or repeating fractional patterns) arise with other rational values of n or c, or other rational increments of x (even if a large irrational initial value of x is used).
Let S(n) = row sums = Sum(k>=1, T(n,k)), then:
S(n) = A077414(n);  S(n)/(n+2) = A000217(n); S(n)/n = A000096(n);
Let Sq(n) = sum of squares of row elements = Sum(k>=1, T(n,k)^2), then:
   Sq(n)/n^2 - 1/n = A058373(n)
Let D(n) = diagonal sums = Sum(k>=1, T(n-k+1, k)) then:
   D(2n) = A131423(n); D(2n-1) = 2/3*n^3 + 1/2*n^2 - 7/6*n;
   D(2n) - D(2n-1) = A000217(n); D(2n+1) - D(2n) = A115067(n);
   D(2n+2) - D(2n)= A056220(n+1);  D(2n+1) - D(2n -1) = A014106(n).
Equals A144204 with the first column of negative ones removed. - Georg Fischer, Jul 26 2023

			Triangle begins as:
0;
1,  3;
2,  5,  8;
3,  7, 11, 15;
4,  9  14, 19, 24;
5, 11, 17, 23, 29, 35;
6, 13, 20, 27, 34, 41, 48;
7, 15, 23, 31, 39, 47, 55, 63;
8, 17, 26, 35, 44, 53, 62, 71, 80;

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Cf. A075362, A144204.

Also note: T(n+1,k) = T(n,k)+ k, and T(n,n) = n^2 - 1.
a(n) = A075362(n)-1; a(n)=i(t+1)-1, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jul 24 2013
T(n, k) = n*k - 1. - Georg Fischer, Jul 26 2023

Simpler name from Georg Fischer, Jul 26 2023

A092393 Triangle read by rows: T(n,k) = (n+k)*binomial(n,k) (for k=0..n-1).

Original entry on oeis.org

1, 2, 6, 3, 12, 15, 4, 20, 36, 28, 5, 30, 70, 80, 45, 6, 42, 120, 180, 150, 66, 7, 56, 189, 350, 385, 252, 91, 8, 72, 280, 616, 840, 728, 392, 120, 9, 90, 396, 1008, 1638, 1764, 1260, 576, 153, 10, 110, 540, 1560, 2940, 3780, 3360, 2040, 810, 190, 11, 132, 715

Offset: 1

Benoit Cloitre, Mar 21 2004

			Triangle starts:
1;
2, 6;
3, 12, 15;
4, 20, 36,  28;
5, 30, 70,  80,  45;
6, 42, 120, 180, 150, 66;
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Cf. A029635.

A092393 := proc(n,k)
    (n+k)*binomial(n,k) ;
end proc:
seq(seq( A092393(n,k),k=0..n-1),n=1..12) ; # R. J. Mathar, Nov 02 2023

A092393row[n_]:=Table[(n+k)Binomial[n,k],{k,0,n-1}];Array[A092393row,10]  (* Paolo Xausa, Nov 02 2023 *)

PARI
```
T(n,k)=binomial(n,k)*(n+k)
```

First column = positive integers;
second column = A002378;
third column = A077414;
main diagonal (i.e., T(n,n) = (n+n)*binomial(n,n) = 2n, which is not included in this sequence) = even integers;
second diagonal = A000384.
Row sums = 1, 8, 30, 88, 230,... = A167667(n)-2n. - R. J. Mathar, Nov 02 2023

A317637 a(n) = n(n+1)(n+3).

Original entry on oeis.org

0, 8, 30, 72, 140, 240, 378, 560, 792, 1080, 1430, 1848, 2340, 2912, 3570, 4320, 5168, 6120, 7182, 8360, 9660, 11088, 12650, 14352, 16200, 18200, 20358, 22680, 25172, 27840, 30690, 33728, 36960, 40392, 44030, 47880, 51948, 56240, 60762, 65520, 70520, 75768, 81270, 87032

Offset: 0

Renzo Remotti, Aug 02 2018

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A077414.

Table[n*(n + 1)*(n + 3), {n, 0, 43}] (* Giovanni Resta, Aug 10 2018 *)

a(n) = 2*A077414(n+1).
Sum_{n>=1} 1/a(n) = 7/36. - Amiram Eldar, Oct 07 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 13/36. - Amiram Eldar, Feb 22 2022

A354968 Triangle read by rows: T(n, k) = nk(n+k)*(n-k)/6.

Original entry on oeis.org

1, 4, 5, 10, 16, 14, 20, 35, 40, 30, 35, 64, 81, 80, 55, 56, 105, 140, 154, 140, 91, 84, 160, 220, 256, 260, 224, 140, 120, 231, 324, 390, 420, 405, 336, 204, 165, 320, 455, 560, 625, 640, 595, 480, 285, 220, 429, 616, 770, 880, 935, 924, 836, 660, 385, 286, 560, 810, 1024

Offset: 2

Ali Sada and Yifan Xie, Jun 14 2022

Given a Pythagorean triple (a,b,c), define S = c^4 - a^4 - b^4. Using Euclid's parameterization (a = 2*n*k, b = n^2 - k^2, c = n^2 + k^2), substituting to get S in terms of n and k gives S = 8*n^2*k^2*((n^2 - k^2))^2, which is a multiple of 288; T(n, k) = sqrt(S/288) = n*k*(n^2 - k^2)/6 = n*k*(n+k)*(n-k)/6.

			Triangle begins:
  n/k   1    2    3    4    5    6    7
  2     1;
  3     4,   5;
  4    10,  16,  14;
  5    20,  35,  40,  30;
  6    35,  64,  81,  80,  55;
  7    56, 105, 140, 154, 140,  91;
  8    84, 160, 220, 256, 260, 224, 140;
  ...
For n = 3, k = 2, a = 5, b = 12, c = 13. T(3, 2) = sqrt((13^4 - 5^4 - 12^4)/288) = 5.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 72.

M. F. Hasler, Table of n, a(n) for n = 2..1000, May 08 2025
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A120070 (b leg), A055096 (c hypotenuse).

Cf. A006414 (row sums), A000292 (column 1), A077414 (column 2), A000330 (diagonal), A107984 (transpose), A210440 (diagonal which begins with 4).

T[n_,k_]:=n*k(n^2-k^2)/6; Table[T[n,k],{n,2,11},{k,n-1}]//Flatten (* Stefano Spezia, Jul 11 2025 *)

apply( {A354968(n, k=0)=k|| k=n-1-(1-n=ceil(sqrt(8*n-7)/2+.5))*(2-n)\2; k*(n-k)*n*(n+k)\6}, [2..66]) \\ M. F. Hasler, May 08 2025

G.f.: x^2*y*(1 + x*y - 4*x^2*y + x^3*y + x^4*y^2)/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Jul 11 2025

A242983 n/2 * (n^3 - 2n^2 - 2n + 5).

Original entry on oeis.org

0, 1, 1, 12, 58, 175, 411, 826, 1492, 2493, 3925, 5896, 8526, 11947, 16303, 21750, 28456, 36601, 46377, 57988, 71650, 87591, 106051, 127282, 151548, 179125, 210301, 245376, 284662, 328483, 377175, 431086, 490576, 556017

Offset: 0

Ralf Stephan, Jun 09 2014

For n>1, number of ways to place two dominoes horizontally on an n X n chessboard.

Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A242856, A242861.

Table[n/2 (n^3-2n^2-2n+5),{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,1,12,58},40] (* Harvey P. Dale, Jul 19 2018 *)

a(n) = A019582(n) + A077414(n-2), n>1.

G.f.: x*(-2*x^3 + 17*x^2 - 4*x + 1) / (1-x)^5.

A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0

Offset: 1

Robert G. Wilson v, Apr 27 2020

\c  0 1  2  3  4   5   6   7   8   9  10  11   12   13    14  15  ...
r\
_0  0 1  0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195  A067998
_1  0 1  1  0 -2  -5  -9 -14 -20 -27 -35 -44  -54  -65  -77  -90  A080956
_2  0 1  2  3  4   5   6   7   8   9  10  11   12   13   14   15  A001477
_3  0 1  3  6 10  15  21  28  36  45  55  66   78   91  105  120  A000217
_4  0 1  4  9 16  25  36  49  64  81 100 121  144  169  196  225  A000290
_5  0 1  5 12 22  35  51  70  92 117 145 176  210  247  287  330  A000326
_6  0 1  6 15 28  45  66  91 120 153 190 231  276  325  378  435  A000384
_7  0 1  7 18 34  55  81 112 148 189 235 286  342  403  469  540  A000566
_8  0 1  8 21 40  65  96 133 176 225 280 341  408  481  560  645  A000567
_9  0 1  9 24 46  75 111 154 204 261 325 396  474  559  651  750  A001106
10  0 1 10 27 52  85 126 175 232 297 370 451  540  637  742  855  A001107
11  0 1 11 30 58  95 141 196 260 333 415 506  606  715  833  960  A051682
12  0 1 12 33 64 105 156 217 288 369 460 561  672  793  924 1065  A051624
13  0 1 13 36 70 115 171 238 316 405 505 616  738  871 1015 1170  A051865
14  0 1 14 39 76 125 186 259 344 441 550 671  804  949 1106 1275  A051866
15  0 1 15 42 82 135 201 280 372 477 595 726  870 1027 1197 1380  A051867
...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.

E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

Cf. A317302 (the same array) but read by ascending antidiagonals.
Sub-arrays: A089000, A139600, A206735;
Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ;
Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,
Number of times k>1 appears: A129654, First occurrence of k: A063778.

Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten

P(r, c) = (r - 2)(c(c-1)/2) + c.

cp's OEIS Frontend

A077415 a(n) = n*(n+2)*(n-2)/3.

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A215862 Number of simple labeled graphs on n+2 nodes with exactly n connected components that are trees or cycles.

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A267370 Partial sums of A140091.

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A213761 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

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A223544 T(n, k) = n*k - 1.

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A092393 Triangle read by rows: T(n,k) = (n+k)*binomial(n,k) (for k=0..n-1).

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A317637 a(n) = n*(n+1)*(n+3).

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A077415 a(n) = n(n+2)(n-2)/3.

A213761 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-5+3h, n>=1, h>=1, and ** = convolution.

A317637 a(n) = n(n+1)(n+3).

A354968 Triangle read by rows: T(n, k) = nk(n+k)*(n-k)/6.

A242983 n/2 * (n^3 - 2n^2 - 2n + 5).