A015237 -id:A015237 - cp's OEIS frontend

A103220 a(n) = n(n+1)(3*n^2+n-1)/6.

0, 1, 13, 58, 170, 395, 791, 1428, 2388, 3765, 5665, 8206, 11518, 15743, 21035, 27560, 35496, 45033, 56373, 69730, 85330, 103411, 124223, 148028, 175100, 205725, 240201, 278838, 321958, 369895, 422995, 481616, 546128, 616913, 694365, 778890

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

Row sums of A103219.
From Bruno Berselli, Dec 10 2010: (Start)
a(n) = n*A002412(n) - Sum_{i=0..n-1} A002412(i). More generally: n^2*(n+1)*(2*d*n-2*d+3)/6 - (Sum_{i=0..n-1} i*(i+1)*(2*d*i-2*d+3))/6 = n * (n+1) * (3*d*n^2-d*n+4*n-2*d+2)/12; in this sequence is d=2.
The inverse binomial transform yields 0, 1, 11, 22, 12, 0, 0 (0 continued). (End)
a(n-1) is also number of ways to place 2 nonattacking semi-queens (see A099152) on an n X n board. - Vaclav Kotesovec, Dec 22 2011
Also, one-half the even-indexed terms of the partial sums of A045947. - J. M. Bergot, Apr 12 2018

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

Cf. A002412, A002418, A039755, A099152, A103219, A015237 (first diffs), A024196.

for(n=0,100,print1((3*n^4+4*n^3-n)/6,","))

CoefficientList[Series[- x (1 + 8 x + 3 x^2) / (x - 1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,13,58,170},40] (* Harvey P. Dale, Jan 23 2016 *)

a(n)=n*(n+1)*(3*n^2+n-1)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1+8*x+3*x^2)/(1-x)^5.
a(n) = Sum_{i=1..n} Sum_{j=1..n} max(i,j)^2. - Enrique Pérez Herrero, Jan 15 2013
a(n) = a(n-1) + (2*n-1)*n^2 with a(0)=0, see A015237. - J. M. Bergot, Jun 10 2017
From Wesley Ivan Hurt, Nov 20 2021: (Start)
a(n) = Sum_{k=1..n} k * C(2*k,2).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). (End)
From Peter Bala, Sep 03 2023: (Start)
a(n) = Sum_{1 <= i <= j <= n} (2*i - 1)*(2*j - 1).
Second subdiagonal of A039755. (End)

A099721 a(n) = n^2(2n+1).

Original entry on oeis.org

0, 3, 20, 63, 144, 275, 468, 735, 1088, 1539, 2100, 2783, 3600, 4563, 5684, 6975, 8448, 10115, 11988, 14079, 16400, 18963, 21780, 24863, 28224, 31875, 35828, 40095, 44688, 49619, 54900, 60543, 66560, 72963, 79764, 86975, 94608, 102675, 111188, 120159, 129600

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 07 2004

For a right triangle with sides of lengths 8*n^3 + 12*n^2 + 8*n + 2, 4*n^4 + 8*n^3 + 4*n^2, and 4*n^4 + 8*n^3 + 12*n^2 + 8*n + 2, dividing the area by the perimeter gives a(n). - J. M. Bergot, Jul 30 2013
This sequence is the difference between the centered icosahedral (or cuboctahedral) numbers (A005902(n)) and the centered octagonal pyramidal numbers (A000447(n+1)). - Peter M. Chema, Jan 09 2016
a(n) is the sum of the integers in the closed interval (n-1)*n to n*(n+1). - J. M. Bergot, Apr 19 2017

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

Row n=3 of A229079.
Cf. A000578, A001093, A011379, A015237, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568, A098547.
Cf. A000290, A000447, A002378, A005408, A005902, A024196.

[n^2*(2*n+1): n in [0..50]]; // Vincenzo Librandi, May 01 2011

A099721 := proc(n) n^2*(2*n+1) ; end proc:
seq(A099721(n),n=0..10) ;

a[n_]:=2*n^3+n^2; (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{4,-6,4,-1},{0,3,20,63},40] (* Harvey P. Dale, Aug 19 2022 *)

a(n) = ceil(sum(i=n^2-(n-1), n^2+(n-1), if(!issquare(4*i+1), (2*i+1+sqrt(4*i+1))/2, 0))); \\ Michel Marcus, Nov 14 2014, after Richard R. Forberg

G.f.: x*(3 + 8*x + x^2)/(x-1)^4.
a(n) = A024196(n) - A024196(n-1). - Philippe Deléham, May 07 2012
a(n) = ceiling(Sum_{i=n^2-(n-1)..n^2+(n-1)} s(i)), for n > 0 and integer i, where s(i) are the real solutions to x = i + sqrt(x), and the summation range excludes the integer solutions which occur where i is an oblong number (A002378). The fractional portion of the summation converges to 2/3 for large n. If s(i) is replaced with i, then the summation equals n^2*(2*n-1) = A015237. - Richard R. Forberg, Oct 15 2014
a(n) = A005902(n) - A000447(n+1). - Peter M. Chema, Jan 09 2016
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 + 4*log(2) - 4.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - Pi - 2*log(2) + 4. (End)
From Elmo R. Oliveira, Aug 08 2025: (Start)
E.g.f.: x*(1 + 2*x)*(3 + x)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A000290(n)*A005408(n). (End)

A199771 Row sums of the triangle in A199332.

Original entry on oeis.org

1, 5, 12, 26, 45, 75, 112, 164, 225, 305, 396, 510, 637, 791, 960, 1160, 1377, 1629, 1900, 2210, 2541, 2915, 3312, 3756, 4225, 4745, 5292, 5894, 6525, 7215, 7936, 8720, 9537, 10421, 11340, 12330, 13357, 14459, 15600, 16820, 18081, 19425, 20812, 22286, 23805

Offset: 1

Reinhard Zumkeller, Nov 23 2011

a(n) = Sum_{k=1..n} A199332(n,k);
a(2*n-1) = A015237(n); a(2*n) = A048395(n);
a(n+1) = A200252(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Haskell
```
a199771  = sum . a199332_row
```

LinearRecurrence[{2,1,-4,1,2,-1},{1,5,12,26,45,75},50] (* Harvey P. Dale, Apr 27 2019 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,2,1,-4,1,2]^(n-1)*[1;5;12;26;45;75])[1,1] \\ Charles R Greathouse IV, Jun 18 2017

G.f.: x*( 1+3*x+x^2+x^3 ) / ((1+x)^2*(x-1)^4). - R. J. Mathar, Nov 24 2011

a(n) = n*(3+2*n^2+4*n+(-1)^n)/8. - R. J. Mathar, Jun 23 2023

A262000 a(n) = n^2(7n - 5)/2.

Original entry on oeis.org

0, 1, 18, 72, 184, 375, 666, 1078, 1632, 2349, 3250, 4356, 5688, 7267, 9114, 11250, 13696, 16473, 19602, 23104, 27000, 31311, 36058, 41262, 46944, 53125, 59826, 67068, 74872, 83259, 92250, 101866, 112128, 123057, 134674, 147000, 160056, 173863, 188442, 203814, 220000

Offset: 0

Bruno Berselli, Sep 08 2015

Also, structured enneagonal prism numbers.

			For n=8, a(8) = 8*(7*0+1)+8*(7*1+1)+8*(7*2+1)+8*(7*3+1)+8*(7*4+1)+8*(7*5+1)+8*(7*6+1)+8*(7*7+1) = 1632.

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. similar sequences with the formula n^2*(k*n - k + 2)/2: A000290 (k=0), A002411 (k=1), A000578 (k=2), A050509 (k=3), A015237 (k=4), A006597 (k=5), A100176 (k=6), this sequence (k=7), A103532 (k=8).

Cf. A001106, A069125.

Magma
```
[n^2*(7*n-5)/2: n in [0..40]];
```

Table[n^2 (7 n - 5)/2, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{0,1,18,72},50] (* Harvey P. Dale, Oct 04 2016 *)

PARI
```
vector(40, n, n--; n^2*(7*n-5)/2)
```
Sage
```
[n^2*(7*n-5)/2 for n in (0..40)]
```

G.f.: x*(1 + 14*x + 6*x^2)/(1 - x)^4.
a(n) = Sum_{i=0..n-1} n*(7*i+1) for n>0, a(0)=0.
a(n+1) + a(-n) = A069125(n+1).
Sum_{i>0} 1/a(i) = 1.082675669875907610300284768825... = (42*(log(14) + 2*(cos(Pi/7)*log(cos(3*Pi/14)) + log(sin(Pi/7))*sin(Pi/14) - log(cos(Pi/14)) * sin(3*Pi/14))) + 21*Pi*tan(3*Pi/14))/75 - Pi^2/15. - Vaclav Kotesovec, Oct 04 2016
From Elmo R. Oliveira, Aug 06 2025: (Start)
E.g.f.: exp(x)*x*(2 + 16*x + 7*x^2)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A272378 a(n) = n(6n^2 - 8*n + 3).

Original entry on oeis.org

0, 1, 22, 99, 268, 565, 1026, 1687, 2584, 3753, 5230, 7051, 9252, 11869, 14938, 18495, 22576, 27217, 32454, 38323, 44860, 52101, 60082, 68839, 78408, 88825, 100126, 112347, 125524, 139693, 154890, 171151, 188512, 207009, 226678, 247555, 269676, 293077

Offset: 0

Vincenzo Librandi, Apr 29 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014 (page 16).
Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

Cf. A000384, A015237, A272379, A272380.

Magma
```
[n*(6*n^2 - 8*n + 3): n in [0..50]];
```

Table[n (6 n^2 - 8 n + 3), {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,1,22,99},40] (* Harvey P. Dale, Dec 29 2017 *)

vector(100, n, n--; n*(6*n^2 - 8*n + 3)) \\ Altug Alkan, Apr 29 2016

for n in range(0,10**3):print(n*(6*n**2-8*n+3),end=", ") # Soumil Mandal, Apr 30 2016

G.f.: x*(1 + 18*x + 17*x^2)/(1 - x)^4.
E.g.f.: x*(1 + 10*x + 6*x^2)*exp(x).
a(n) = n*A080859(n+1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), for n>3.
See page 7 in Brent's paper:
a(n) = n^2*A000384(n) - n*(n-1)*A000384(n-1).
A272379(n) = n^2*a(n) - n*(n-1)*a(n-1).
From Peter Bala, Jan 30 2019: (Start)
Let a(n,x) = Product_{k = 0..n} (x - k)/(x + k). Then for positive integer x we have x^2*(6*x^2 - 8*x + 3) = Sum_{n >= 0} ((n+1)^7 + n^7)*a(n,x) and x*(6*x^2 - 8*x + 3) = Sum_{n >= 0} ((n+1)^6 - n^6)*a(n,x). Both identities are also valid for complex x in the half-plane Re(x) > 7/2. See the Bala link in A036970. Cf. A272379. (End)

A256141 Square array read by antidiagonals upwards: T(n,k), n>=0, k>=0, in which row n lists the partial sums of the n-th row of the square array of A256140.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 7, 9, 5, 1, 1, 6, 9, 16, 11, 6, 1, 1, 7, 11, 25, 19, 15, 7, 1, 1, 8, 13, 36, 29, 28, 19, 8, 1, 1, 9, 15, 49, 41, 45, 37, 27, 9, 1, 1, 10, 17, 64, 55, 66, 61, 64, 29, 10, 1, 1, 11, 19, 81, 71, 91, 91, 125, 67, 33, 11, 1, 1, 12, 21, 100, 89, 120, 127, 216, 129, 76, 37, 12, 1

Offset: 0

Omar E. Pol, Mar 16 2015

Questions:
Is also A130667 a row of this square array?
Is also A116522 a row of this square array?
Is also A116526 a row of this square array?
Is also A116525 a row of this square array?
Is also A116524 a row of this square array?

			The corner of the square array with the first 15 terms of the first 12 rows looks like this:
--------------------------------------------------------------------------
A000012: 1, 1, 1,  1,  1,  1,  1,   1,   1,   1,   1,   1,   1,   1,   1
A000027: 1, 2, 3,  4,  5,  6,  7,   8,   9,  10,  11,  12,  13,  14,  15
A006046: 1, 3, 5,  9, 11, 15, 19,  27,  29,  33,  37,  45,  49,  57,  65
A130665: 1, 4, 7, 16, 19, 28, 37,  64,  67,  76,  85, 112, 121, 148, 175
A116520: 1, 5, 9, 25, 29, 45, 61, 125, 129, 145, 161, 225, 241, 305, 369
A130667? 1, 6,11, 36, 41, 66, 91, 216, 221, 246, 271, 396, 421, 546, 671
A116522? 1, 7,13, 49, 55, 91,127, 343, 349, 385, 421, 637, 673, 889,1105
A161342: 1, 8,15, 64, 71,120,169, 512, 519, 568, 617, 960,1009,1352,1695
A116526? 1, 9,17, 81, 89,153,217, 729, 737, 801, 865,1377,1441,1953,2465
.......: 1,10,19,100,109,190,271,1000,1009,1090,1171,1900,1981,2710,3439
A116525? 1,11,21,121,131,231,331,1331,1341,1441,1541,2541,2641,3641,4641
.......: 1,12,23,144,155,276,397,1728,1739,1860,1981,3312,3422,4764,6095

Index entries for sequences related to cellular automata

Cf. A000120, A255740, A255741, A256140.
First five rows are A000012, A000027, A006046, A130665, A116520. Row 7 is A161342.
First eight columns are A000012, A000027, A005408, A000290, A028387, A000384, A003215, A000578. Column 9 is A081437. Column 11 is A015237. Columns 13-15 are A005915, A005917, A000583.

A141680 Triangle read by rows: T(n,m) = (n/m)*binomial(n,m) if m divides n, otherwise 0.

Original entry on oeis.org

1, 4, 1, 9, 0, 1, 16, 12, 0, 1, 25, 0, 0, 0, 1, 36, 45, 40, 0, 0, 1, 49, 0, 0, 0, 0, 0, 1, 64, 112, 0, 140, 0, 0, 0, 1, 81, 0, 252, 0, 0, 0, 0, 0, 1, 100, 225, 0, 0, 504, 0, 0, 0, 0, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 07 2008

Row sums are: 1, 5, 10, 29, 26, 122, 50, 317, 334, 830, ... A105862.

			1;
4, 1;
9, 0, 1;
16, 12, 0, 1;
25, 0, 0, 0, 1;
36, 45, 40, 0, 0, 1;
49, 0, 0, 0, 0, 0, 1;
64, 112, 0, 140, 0, 0, 0, 1;
81, 0, 252, 0, 0, 0, 0, 0, 1;
100, 225, 0, 0, 504, 0, 0, 0, 0, 1;

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

Cf. A015237, A105862, A126988.

t[n_, m_] = If[Mod[n, m] == 0, n/m, 0]*Binomial[n, m]; Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[%]

T(n,m) = A126988(n,m)*binomial(n,m).

T(n,1) = n^2. T(n,n) = 1. T(2n,2) = A015237(n).

A143218 Triangle read by rows, A127775 * A000012 * A127775; 1<=k<=n.

Original entry on oeis.org

1, 3, 9, 5, 15, 25, 7, 21, 35, 49, 9, 27, 45, 63, 81, 11, 33, 55, 77, 99, 121, 13, 39, 65, 91, 117, 143, 169, 15, 45, 75, 105, 135, 165, 195, 225, 17, 51, 85, 119, 153, 187, 221, 255, 289, 19, 57, 95, 133, 171, 209, 247, 285, 323, 361, 21, 63, 105, 147, 189, 231, 273, 315, 357, 399, 441

Offset: 1

Gary W. Adamson, Jul 30 2008

			First few rows of the triangle =
   1;
   3,  9;
   5, 15, 25;
   7, 21, 35, 49;
   9, 27, 45, 63,  81;
  11, 33, 55, 77,  99, 121;
  13, 39, 65, 91, 117, 143, 169;
  ...
T(5,3) = 45 = 9*5 = (2*5 - 1) * (2*3 - 1).

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000012, A005408, A014634, A015237 (row sums).

Cf. A016754, A024598, A033567, A127775, A131507.

[(2*n-1)*(2*k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 12 2022

Table[(2*k-1)*(2*n-1), {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 12 2022 *)

flatten([[(2*n-1)*(2*k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 12 2022

Triangle read by rows, A127775 * A000012 * A127775.
T(n, k) = (2*n - 1) * (2*k - 1), 1<=k<=n.
Sum_{k=1..n} T(n, k) = A015237(n) = n^2 * (2*n-1).
From G. C. Greubel, Jul 12 2022: (Start)
T(n, k) = A131507(n,k) * A127775(n,k).
T(n, n) = A016754(n-1) = (2*n-1)^2, n >= 1.
T(2*n-1, n) = A014634(n-1), n >= 1.
T(2*n-2, n-1) = A033567(n-1), n >= 2.
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024598(n), n >= 1. (End)

A280391 Number of 2 X 2 matrices with all elements in {0,...,n} with permanent = determinant * n.

Original entry on oeis.org

1, 12, 25, 57, 81, 141, 169, 259, 297, 413, 441, 621, 625, 825, 873, 1079, 1089, 1403, 1369, 1739, 1729, 2021, 2025, 2507, 2433, 2859, 2905, 3301, 3249, 4029, 3721, 4509, 4305, 4793, 4989, 5551, 5329, 6027, 6025, 6807, 6561, 7917, 7225, 8357, 8121, 8677, 8649, 9843, 9481, 10889

Offset: 0

Indranil Ghosh, Jan 02 2017

nonn

All the values except a(1) are odd.
From Robert Israel, Jan 02 2017: (Start)
Number of solutions to (n+1)*x*y = (n-1)*z*w for x,y,z,w in [0..n].
a(n) >= (2n+1)^2, with equality if n+1 is an odd prime. (End)

Robert Israel and Indranil Ghosh, Table of n, a(n) for n = 0..1600 (n = 0..200 from Indranil Ghosh)

Cf. A280321 (Number of 2 X 2 matrices with all elements in {0,..,n} with permanent*n = determinant).
Cf. A015237 (Number of 2 X 2 matrices having all elements in {0..n} with determinant = permanent).
Cf. A016754 (Number of 2 X 2 matrices having all elements in {0..n} with determinant =2* permanent).
Cf. A280364 (Number of 2 X 2 matrices having all elements in {0..n} with determinant^n = permanent).

g:= proc(r,n) if r = 0 then 2*n+1 else nops(select(t -> t <= n and r <= t*n, numtheory:-divisors(r))) fi end proc:
f:= proc(n) local c;
    if n::even then (2*n+1)^2 + add(g((n+1)*c,n)*g((n-1)*c,n), c=1..n-1)
    else (2*n+1)^2 + add(g((n+1)/2*c,n) * g((n-1)/2*c,n), c=1..2*n-1)
    fi
end proc:
map(f, [$0..100]); # Robert Israel, Jan 02 2017

g[r_, n_] := If[r == 0, 2n + 1, Length[Select[Divisors[r], # <= n && r <= # n&]]];
f[n_] := If[EvenQ[n], (2n + 1)^2 + Sum[g[(n + 1)c, n] g[(n - 1)c, n], {c, 1, n - 1}], (2n + 1)^2 + Sum[g[(n + 1)/2 c, n] g[(n - 1)/2 c, n], {c, 1, 2n - 1}]];
f /@ Range[0, 100] (* Jean-François Alcover, Jul 29 2020, after Robert Israel *)

def t(n):
    s=0
    for a in range(0,n+1):
        for b in range(0,n+1):
            for c in range(0,n+1):
                for d in range(0,n+1):
                    if (a*d-b*c)*n==(a*d+b*c):
                        s+=1
    return s
for i in range(0,201):
    print(t(i))

A247327 Triangle read by rows: T(n,k) = sum of k-th row of n X n square filled with odd numbers 1 through 2*n^2-1 reading across rows left-to-right.

Original entry on oeis.org

1, 4, 12, 9, 27, 45, 16, 48, 80, 112, 25, 75, 125, 175, 225, 36, 108, 180, 252, 324, 396, 49, 147, 245, 343, 441, 539, 637, 64, 192, 320, 448, 576, 704, 832, 960, 81, 243, 405, 567, 729, 891, 1053, 1215, 1377, 100, 300, 500, 700, 900, 1100, 1300, 1500, 1700, 1900, 121, 363

Offset: 1

Kival Ngaokrajang, Sep 13 2014

See illustration in links. Column c(k) = (2*k - 1)*n^2. Diagonal d(m) = (2*n - 2*m + 1)*n^2.

			Triangle begins:
  1
  4   12
  9   27  45
  16  48  80 112
  25  75 125 175 225
  36 108 180 252 324 396
  49 147 245 343 441 539 637

Kival Ngaokrajang, Illustration of initial terms

Column: c(1) = A000290, c(2) = A033428, c(3) = A033429.
Diagonal: d(1) = A015237, d(2) = A015238, d(3) = A015240.
Rows sum: A000538.
Cf. A241016.

trg(nn) = {for (n=1, nn, mm = matrix(n, n, i, j, (2*j-1) + (2*n)*(i-1)); for (i=1, n, print1(sum(j=1, n, mm[i, j]), ", ");); print(););} \\ Michel Marcus, Sep 15 2014

cp's OEIS Frontend

A103220 a(n) = n*(n+1)*(3*n^2+n-1)/6.

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A099721 a(n) = n^2*(2*n+1).

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Magma

Maple

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A199771 Row sums of the triangle in A199332.

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Haskell

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A262000 a(n) = n^2*(7*n - 5)/2.

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Magma

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Sage

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A272378 a(n) = n*(6*n^2 - 8*n + 3).

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Magma

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A256141 Square array read by antidiagonals upwards: T(n,k), n>=0, k>=0, in which row n lists the partial sums of the n-th row of the square array of A256140.

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A141680 Triangle read by rows: T(n,m) = (n/m)*binomial(n,m) if m divides n, otherwise 0.

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A103220 a(n) = n(n+1)(3*n^2+n-1)/6.

A099721 a(n) = n^2(2n+1).

A262000 a(n) = n^2(7n - 5)/2.

A272378 a(n) = n(6n^2 - 8*n + 3).