A006000 -id:A006000 - cp's OEIS frontend

A323956 Triangle read by rows: T(n, k) = 1 + n * (n - k) for 1 <= k <= n.

1, 3, 1, 7, 4, 1, 13, 9, 5, 1, 21, 16, 11, 6, 1, 31, 25, 19, 13, 7, 1, 43, 36, 29, 22, 15, 8, 1, 57, 49, 41, 33, 25, 17, 9, 1, 73, 64, 55, 46, 37, 28, 19, 10, 1, 91, 81, 71, 61, 51, 41, 31, 21, 11, 1, 111, 100, 89, 78, 67, 56, 45, 34, 23, 12, 1

Offset: 1

Gus Wiseman, Feb 10 2019

			Triangle begins:
  n\k:   1   2   3   4   5   6   7   8   9  10  11  12
  ====================================================
    1:   1
    2:   3   1
    3:   7   4   1
    4:  13   9   5   1
    5:  21  16  11   6   1
    6:  31  25  19  13   7   1
    7:  43  36  29  22  15   8   1
    8:  57  49  41  33  25  17   9   1
    9:  73  64  55  46  37  28  19  10   1
   10:  91  81  71  61  51  41  31  21  11   1
   11: 111 100  89  78  67  56  45  34  23  12   1
   12: 133 121 109  97  85  73  61  49  37  25  13   1
  etc.

G. C. Greubel, Rows n = 1..100 of triangle, flattened

First column is A002061. Second column is A000290. Third column is A028387.

Cf. A000126, A001610, A001644, A006000, A169985, A306351, A323952, A323955.

[[1+n*(n-k): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Apr 22 2019

Mathematica
```
Table[1+n*(n-k),{n,12},{k,n}]//Flatten
```

{T(n,k) = 1+n*(n-k)}; \\ G. C. Greubel, Apr 22 2019

[[1+n*(n-k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Apr 22 2019

From Werner Schulte, Feb 12 2019: (Start)
G.f.: Sum_{n>0,k=1..n} T(n,k)*x^k*t^n = x*t*((1-t+2*t^2)*(1-x*t) + (1-t)*t)/((1-t)^3*(1-x*t)^2).
Row sums: Sum_{k=1..n} T(n,k) = A006000(n-1) for n > 0.
Recurrence: T(n,k) = T(n,k-1) - n for 1 < k <= n with initial values T(n,1) = n^2-n+1 for n > 0.
Recurrence: T(n,k) = T(n-1,k) + 2*n-k-1 for 1 <= k < n with initial values T(n,n) = 1 for n > 0.
(End)

A218152 a(n) = 1 + n + ((n-1)*n^2)/2.

Original entry on oeis.org

1, 2, 5, 13, 29, 56, 97, 155, 233, 334, 461, 617, 805, 1028, 1289, 1591, 1937, 2330, 2773, 3269, 3821, 4432, 5105, 5843, 6649, 7526, 8477, 9505, 10613, 11804, 13081, 14447, 15905, 17458, 19109, 20861, 22717, 24680, 26753, 28939, 31241, 33662, 36205, 38873

Offset: 0

Mokhtar Mohamed, Oct 24 2012

a(n) = sum(i=0,1,2,...k) d(i)*C(n,i), d(0)=a(0), C(n,i)=0 for all i > n. I would introduce the arithmetic-arithmetic sequence which is defined as the sequence of finite differences, that is, with k consecutive rows of differences, whose first terms are d(1), d(2), d(3),..., d(k), the last row (k-th row) being of a constant difference. Here, it is submitted a special case of the above mentioned sequence with k=3, d(0)=d(1)=1,  d(2)=2, d(3)=3.
This sequence is not in Comtet. - T. D. Noe, Nov 16 2012
a(n) appears to be the number of configurations of n equilateral triangles that are allowed to have common vertices, where A002061(n) gives the number of connected configurations and A060354(n) is the number of configurations consisting of several pieces. - Anton Zakharov, May 13 2018

			for n=5, a(5) = 1+5+(4*25)/2 = 1+5+100/2 = 1+5+50 = 56.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72.

Colin Barker, Table of n, a(n) for n = 0..1000
Anton Zakharov, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000027, A000124, A000125, A060354, A002061.

Table[1+n+((n-1)n^2)/2,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,2,5,13},50] (* Harvey P. Dale, May 04 2023 *)

Vec((1 - 2*x + 3*x^2 + x^3) / (1 - x)^4 + O(x^40)) \\ Colin Barker, May 13 2018

a(n) = a(n-1)+(4-5*n+3*n^2)/2 for n > 0 and a(0)=1.
a(n) = A006000(n-1)+1 for n > 0. - Antti Karttunen, Oct 24 2012
a(n) = A060354(n) + A002061(n). - Anton Zakharov, May 13 2018
G.f.: (x^3+3*x^2-2*x+1)/(x-1)^4. - Alois P. Heinz, May 13 2018
From Colin Barker, May 13 2018: (Start)
a(n) = (2 + 2*n - n^2 + n^3) / 2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
(End)

Corrected and edited by Mokhtar Mohamed, Nov 17 2012

Missing term 1937 inserted by Alois P. Heinz, Jun 11 2017

A249973 Positive integers A when the positive roots of r^2 = Ar + B are listed in increasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 1, 2, 3, 4, 1, 2, 3, 1, 4, 2, 1, 3, 2, 4, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 5, 3, 1, 4, 2, 1, 3, 5, 2, 4, 1, 3, 2, 5, 1, 4, 3

Offset: 1

Kerry Mitchell, Nov 09 2014

Generalize the Fibonacci sequence recurrence equation as: F_(n+1) = A*F_n + B*F_(n-1), where A and B are positive integers. As n goes to infinity, the ratio F_n / F_(n-1) approaches the positive real number r = (A + sqrt(A*A + 4B))/2. This sequence gives the A values in increasing order of r.
In case of a tie in r values, then sort in increasing order of sqrt(A*A + B*B).
This A sequence appears to be the ordinal transform of the B sequence (A249974) and vice versa. The associative arrays of A and B are transposes. The first row of A's associative array seems to be A006000.
For the A and B values leading to a positive integer limit r see a comment in A063929. - Wolfdieter Lang, Jan 12 2015

			a(6) = 2 because the 6th smallest value of r (approximately 2.732050808) is that for A=2, B=2.

Cf. A249974, A006000.

PARI
```
\\ see A249974
```

Edited. - Wolfdieter Lang, Jan 11 2015

A251630 Column sums of the n X n square array filled with numbers from 1 to n^2, row by row, from left to right.

Original entry on oeis.org

1, 4, 6, 12, 15, 18, 28, 32, 36, 40, 55, 60, 65, 70, 75, 96, 102, 108, 114, 120, 126, 154, 161, 168, 175, 182, 189, 196, 232, 240, 248, 256, 264, 272, 280, 288, 333, 342, 351, 360, 369, 378, 387, 396, 405, 460, 470, 480, 490, 500, 510, 520, 530

Offset: 1

Wolfdieter Lang, Dec 09 2014

This triangle has been considered by Kival Ngaokrajang as a companion of A241016. See the link given there, the second triangle.

			The n=4 square array is:
1   2  3  4
5   6  7  8
9  10 11 12
13 14 15 16
and the column sums are 28 32 36 40, which appear
in row n=4 of the triangle T.
The triangle T(n,k) begins:
n\k   1   2   3   4   5   6   7   8   9  10 ...
1:    1
2:    4   6
3:   12  15  18
4:   28  32  36  40
5:   55  60  65  70  75
6:   96 102 108 114 120 126
7:  154 161 168 175 182 189 196
8:  232 240 248 256 264 272 280 288
9:  333 342 351 360 369 378 387 396 405
10: 460 470 480 490 500 510 520 530 540 550
...

Cf. A002411 (main diagonal), A006000 (column k=1), A241016.

T(n, k) = sum(n*(j-1)+ k, j=1..n), n >= k >= 1.

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

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

A329523 a(n) = n * (binomial(n + 1, 3) + 1).

Original entry on oeis.org

0, 1, 4, 15, 44, 105, 216, 399, 680, 1089, 1660, 2431, 3444, 4745, 6384, 8415, 10896, 13889, 17460, 21679, 26620, 32361, 38984, 46575, 55224, 65025, 76076, 88479, 102340, 117769, 134880, 153791, 174624, 197505, 222564, 249935, 279756, 312169, 347320, 385359, 426440

Offset: 0

Ilya Gutkovskiy, Nov 15 2019

The n-th centered n-gonal pyramidal number.

			Square array begins:
  (0), 1,  2,   3,   4,    5,  ... A001477
   0, (1), 3,   7,  14,   25,  ... A004006
   0,  1, (4), 11,  24,   45,  ... A006527
   0,  1,  5, (15), 34,   65,  ... A006003 (partial sums of A005448)
   0,  1,  6,  19, (44),  85,  ... A005900 (partial sums of A001844)
   0,  1,  7,  23,  54, (105), ... A004068 (partial sums of A005891)
...
This sequence is the main diagonal of the array.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), 142.

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000292, A000330, A002415, A002417, A006000, A006484, A008911, A050407, A060354, A100119, A188475 (first differences).

[ n*(Binomial(n+1,3)+1):n in [0..40]]; // Marius A. Burtea, Nov 15 2019

R:=PowerSeriesRing(Integers(), 41); [0] cat Coefficients(R!(x*(1-x+5*x^2-x^3)/(1-x)^5)); // Marius A. Burtea, Nov 15 2019

Table[n (Binomial[n + 1, 3] + 1), {n, 0, 40}]
nmax = 40; CoefficientList[Series[x (1 - x + 5 x^2 - x^3)/(1 - x)^5, {x, 0, nmax}], x]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 4, 15, 44}, 41]

G.f.: x * (1 - x + 5*x^2 - x^3) / (1 - x)^5.
E.g.f.: exp(x) * x * (1 + x + x^2 + x^3 / 6).
a(n) = n * (n + 2) * (n^2 - 2*n + 3) / 6.
a(n) = n * (A000292(n-1) + 1).
a(n) = n + 2 * Sum_{k=1..n} A000330(k-1).
a(n) + a(-n) = 4 * A002415(n).

A361474 a(n) = 1binomial(n,2) + 3binomial(n,3) + 6binomial(n,4) + 10binomial(n,5).

Original entry on oeis.org

0, 0, 1, 6, 24, 80, 225, 546, 1176, 2304, 4185, 7150, 11616, 18096, 27209, 39690, 56400, 78336, 106641, 142614, 187720, 243600, 312081, 395186, 495144, 614400, 755625, 921726, 1115856, 1341424, 1602105, 1901850, 2244896, 2635776, 3079329, 3580710, 4145400, 4779216, 5488321, 6279234

Offset: 0

Enrique Navarrete, Mar 13 2023

a(n) is the number of ordered set partitions of an n-set into 2 sets such that the first set has either 3, 2, 1 or no elements, the second set has no restrictions, and two elements are selected from the second set.

Note the coefficients 1,3,6,10 in a(n) are triangular numbers (in accordance with the selection of two elements from the second set).

			The 546 set partitions for n=7 are the following (where the 2 elements selected from the second set are in parentheses):
   { }, {(1),(2),3,4,5,6,7}  (21 of these);
   {1}, {(2),(3),4,5,6,7}    (105 of these);
   {1,2}, {(3),(4),5,6,7}    (210 of these);
   {1,2,3}, {(4),(5),6,7}    (210 of these).

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

Cf. A006000, A361099.

a[n_] := Total[Binomial[n, Range[2, 5]]*{1, 3, 6, 10}]; Array[a, 40, 0] (* Amiram Eldar, Mar 28 2023 *)

a(n) = binomial(n,2) + 3*binomial(n,3) + 6*binomial(n,4) + 10*binomial(n,5); \\ Michel Marcus, Mar 16 2023

def A361474(n): return n*(n*(n*(n*(n - 7) + 23) - 29) + 12)//12 # Chai Wah Wu, Apr 16 2023

E.g.f.: (1 + x + x^2/2 + x^3/6)*x^2/2*exp(x).

O.g.f.: x^2*(1 + 3*x^2 + 6*x^3)/(1 - x)^6. - Stefano Spezia, Mar 16 2023

A364843 Integers are repeated in runs of 1, 2, 3, ... Each new integer (following a run) is given the value of its sequence index value.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 7, 7, 7, 7, 11, 11, 11, 11, 11, 16, 16, 16, 16, 16, 16, 22, 22, 22, 22, 22, 22, 22, 29, 29, 29, 29, 29, 29, 29, 29, 37, 37, 37, 37, 37, 37, 37, 37, 37, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56

Offset: 1

Peter Woodward, Aug 10 2023

Omitting repeats yields the triangular numbers plus 1 sequence A000124.

			Illustrated as a triangle begins:
   1;
   2,  2;
   4,  4,  4;
   7,  7,  7,  7;
  11, 11, 11, 11, 11;
  16, 16, 16, 16, 16, 16;
  22, 22, 22, 22, 22, 22, 22;
  ...

Cf. A000124, A002024, A006528.

Row sums give A006000(n-1).

T:= (n, k)-> n*(n-1)/2+1:
seq(seq(T(n,k), k=1..n), n=1..11);  # Alois P. Heinz, Aug 31 2023

a(n) = my(t=(sqrtint(8*n-1)-1)\2); t*(t+1)/2+1 \\ Thomas Scheuerle, Aug 10 2023

from math import isqrt
def A364843(n): return ((t:=isqrt((n<<3)-1)-1>>1)*(t+1)>>1)+1 # Chai Wah Wu, Sep 15 2023

G.f.: x*y*(1 + 2*x^4*y^2 - x*(1 + y) - 2*x^3*y*(1 + y) + x^2*(1 + y + y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Sep 02 2023

Sum_{k=1..n} k = T(n,k) = A006528(n). - Alois P. Heinz, Sep 15 2023

A330601 Array T read by antidiagonals: T(m,n) is the number of lattice walks from (0,0) to (m,n) using one step from {(3,0), (2,1), (1,2), (0,3)} and all other steps from {(1,0), (0,1)}.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 4, 4, 4, 2, 3, 9, 12, 12, 9, 3, 4, 16, 28, 32, 28, 16, 4, 5, 25, 55, 75, 75, 55, 25, 5, 6, 36, 96, 156, 180, 156, 96, 36, 6, 7, 49, 154, 294, 392, 392, 294, 154, 49, 7, 8, 64, 232, 512, 784, 896, 784, 512, 232, 64, 8, 9, 81, 333, 837, 1458, 1890, 1890, 1458, 837, 333, 81, 9

Offset: 0

Steven Klee, Dec 19 2019

			For (m,n) = (3,1), there are T(3,1) = 4 paths:
(3,0), (0,1)
(0,1), (3,0)
(2,1), (1,0)
(1,0), (2,1).
Array T(m,n) begins
n/m 0   1    2     3     4      5      6      7       8       9
0   0   0    0     1     2      3      4      5       6       7
1   0   0    1     4     9     16     25     36      49      64
2   0   1    4    12    28     55     96    154     232     333
3   1   4   12    32    75    156    294    512     837    1300
4   2   9   28    75   180    392    784   1458    2550    4235
5   3  16   55   156   392    896   1890   3720    6897   12144
6   4  25   96   294   784   1890   4200   8712   17028   31603
7   5  36  154   512  1458   3720   8712  19008   39039   76076
8   6  49  232   837  2550   6897  17028  39039   84084  171600
9   7  64  333  1300  4235  12144  31603  76076  171600  366080

T(m,0) is A000027 for m >= 2.
T(m,1) is A000290 for m >= 1.
T(m,2) is A006000.

def T(m,n):
    return (m+n-2)*(binomial(m+n-2, m) + binomial(m+n-2, n))

T(m,n) = (m+n-2)*(binomial(m+n-2,m) + binomial(m+n-2,n)).

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

A323956 Triangle read by rows: T(n, k) = 1 + n * (n - k) for 1 <= k <= n.

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A218152 a(n) = 1 + n + ((n-1)*n^2)/2.

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A249973 Positive integers A when the positive roots of r^2 = Ar + B are listed in increasing order.

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A251630 Column sums of the n X n square array filled with numbers from 1 to n^2, row by row, from left to right.

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A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

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A329523 a(n) = n * (binomial(n + 1, 3) + 1).

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A361474 a(n) = 1*binomial(n,2) + 3*binomial(n,3) + 6*binomial(n,4) + 10*binomial(n,5).

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A361474 a(n) = 1binomial(n,2) + 3binomial(n,3) + 6binomial(n,4) + 10binomial(n,5).