A002411 -id:A002411 - cp's OEIS frontend

A261720 Array of pyramidal (3-dimensional figurate numbers) read by antidiagonals.

1, 1, 4, 1, 5, 10, 1, 6, 14, 20, 1, 7, 18, 30, 35, 1, 8, 22, 40, 55, 56, 1, 9, 26, 50, 75, 91, 84, 1, 10, 30, 60, 95, 126, 140, 120, 1, 11, 34, 70, 115, 161, 196, 204, 165, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286

Offset: 1

Gary W. Adamson, Aug 29 2015

First few sequences in the array:
  1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
  1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
  1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
  1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
  1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
  1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
  1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584
  ...
The corresponding bases to rows are: Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, ...

			Row 2: (1, 5, 14, 30, 55, ...) = (1, 4, 10, 20, 35, ...) + (0, 1, 4, 10, 20, 35, ...).
(1, 7, 22, 50, ...) is the binomial transform of (1, 6, 9, 4, 0, 0, 0, ...) 3rd row in Pascal's triangle (1,4) followed by zeros. (1, 7, 22, 50, ...) is the third partial sum of (1, 4, 4, 4, ...).

Albert H. Beiler, "Recreations in the Theory of Numbers"; Dover, 1966, p. 194.

Cf. A000292, A000330, A002411, A002412, A002413, A002414, A007584, A220084.

Similar to A080851 but without row n=0.

T[n_,k_]:=k(k+1)((k-1)n+3)/6; Flatten[Table[T[n-k+1,k],{n,11},{k,n}]] (* Stefano Spezia, Aug 15 2024 *)

T(n,k) = A080851(n,k).
Given: first sequence in the array is A000292: (1, 4, 10, 20, 35, ...) Subsequent rows are generated by adding (0, 1, 4, 10, 20, 35, ...) to the current row.
n-th row is the binomial transform of row 3 in Pascal's triangle (1,n) followed by zeros.  Alternatively, begin with (1, 4, 10, 20, ...) being the binomial transform of (1, 3, 3, 1, 0, 0, 0, ...). Add (0, 1, 2, 1, 0, 0, 0, ...) to the latter to obtain the inverse binomial transform of the next row: (1, 5, 14, 30, 55,..); then repeat the operation.
The row starting (1, N, ...) is the 3rd partial sum of (1, (N-3), (N-3), (N-3), ...).
From Stefano Spezia, Aug 15 2024: (Start)
T(n,k) = k*(k + 1)*((k - 1)*n + 3)/6.
G.f. as array: x*y*(1 + x*(y - 1))/((1 - x)^2*(1 - y)^4).
E.g.f. as array: exp(y)*y*(exp(x)*(6 + 3*(1 + x)*y + x*y^2) - 3*(2 + y))/6. (End)

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)

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)

A285522 Array read by antidiagonals: T(m,n) = number of circulant digraphs up to Cayley isomorphism on n vertices with edges colored according to step value using a maximum of m-1 colors.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 6, 4, 1, 1, 6, 18, 10, 5, 1, 1, 20, 24, 40, 15, 6, 1, 1, 14, 135, 70, 75, 21, 7, 1, 1, 48, 130, 544, 165, 126, 28, 8, 1, 1, 52, 648, 700, 1625, 336, 196, 36, 9, 1, 1, 140, 1137, 4480, 2635, 3996, 616, 288, 45, 10, 1

Offset: 1

Andrew Howroyd, Apr 20 2017

For the base case of m=2 the sequence counts circulant digraphs up to Cayley isomorphism. Two circulant graphs are Cayley isomorphic if there is a d, which is necessarily prime to n, that transforms through multiplication modulo n the step values of one graph into those of the other. For squarefree n this is the only way that two circulant graphs can be isomorphic. (See Liskovets reference for a proof.)
Alternatively, the number of mappings with domain {1..n-1} and codomain {1..m} up to equivalence. Mappings A and B are equivalent if there is a d, prime to n, such that A(i) = B(i*d mod n) for i in {1..n-1}. This sequence differs from A132191 only in that sequence also includes 0 in the domain which introduces an extra factor of m into the results since zero multiplied by anything is zero.
All column sequences are polynomials of order n-1 and these are the cycle index polynomials.
This sequence is also related to A075195(n, m) which counts necklaces and A285548(m, n) which is the sequence described in the Titsworth reference. In particular, A075195 is the analogous array with equivalence determined through the additive group instead of by multiplication whereas A285548 allows for both addition and multiplication.

			Table starts:
\n  1 2  3   4    5    6    7     8      9      10
m\ ---------------------------------------------------
1 | 1 1  1   1    1    1    1     1      1       1 ...
2 | 1 2  3   6    6   20   14    48     52     140 ...
3 | 1 3  6  18   24  135  130   648   1137    4995 ...
4 | 1 4 10  40   70  544  700  4480  11056   65824 ...
5 | 1 5 15  75  165 1625 2635 20625  65425  489125 ...
6 | 1 6 21 126  336 3996 7826 72576 280596 2521476 ...
...
Case n=10:
Only 1, 3, 7, 9 are prime to 10.
Multiplication modulo 10 is described by the following multiplication table.
  1, 2, 3, 4, 5, 6, 7, 8, 9  => (1)(2)(3)(4)(5)(6)(7)(8)(9) => m^9
  3, 6, 9, 2, 5, 8, 1, 4, 7  => (1397)(2684)(5)             => m^3
  7, 4, 1, 8, 5, 2, 9, 6, 3  => (1793)(2486)(5)             => m^3
  9, 8, 7, 6, 5, 4, 3, 2, 1  => (19)(28)(37)(46)(5)         => m^5
Each row of the multiplication table can be viewed as a permutation and together these form a commutative group on 4 elements. In this case the group is isomorphic to the cyclic group C_4. Each permutation can be represented in cycle notation. (shown above to the right of the corresponding multiplication table row). In order to count the equivalence classes using Polya's enumeration theorem only the number of cycles in each permutation is needed.
This gives the cycle index polynomial (1/4)*(m^9 + m^5 + 2*m^3). Putting m = 1..4 gives 1, 140, 4995, 65824.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
V. A. Liskovets and R. Poeschel, On the enumeration of circulant graphs of prime-power and squarefree orders.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
Eric Weisstein's World of Mathematics, Circulant Graph.
Wikipedia, Polya enumeration theorem.

Row 2 is A056391.
Columns include A002411, A006528, A168178, A006565, A282613, A054624.
Cf. A132191, A075195, A285548, A002729.

A132191[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #] &], 0], {k, 1, n}];
T[m_, n_] := A132191[m, n]/m;
Table[T[m - n + 1, n], {m, 1, 11}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Jun 06 2017 *)

a(n,x)=sum(k=1, n, if(gcd(k, n)==1, x^(sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d)))-1), 0))/eulerphi(n);
for(m=1, 6, for(n=1, 10, print1( a(n,m), ", ") ); print(); );

T(m, n) = A132191(m, n) / m.

A319184 Numbers that are sums of consecutive pentagonal numbers.

Original entry on oeis.org

0, 1, 5, 6, 12, 17, 18, 22, 34, 35, 39, 40, 51, 57, 69, 70, 74, 75, 86, 92, 108, 117, 120, 121, 125, 126, 145, 156, 162, 176, 178, 190, 195, 196, 209, 210, 213, 247, 248, 262, 270, 279, 282, 287, 288, 321, 330, 354, 365, 376, 386, 387, 399, 404, 405, 424, 425, 438, 457, 475

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000326, A002411, A034705, A034706, A319185, A322636, A322637.

anmax = 1000; nmax = Floor[Sqrt[2*anmax/3]] + 1; Select[Union[Flatten[Table[Sum[k*(3*k-1)/2, {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)
Module[{nn=20,pn},pn=PolygonalNumber[5,Range[0,nn]];Take[Union[Flatten[Table[Total/@Partition[pn,d,1],{d,nn}]]],60]] (* Harvey P. Dale, Jun 22 2025 *)

A014799 Squares of odd pentagonal pyramidal numbers.

Original entry on oeis.org

1, 5625, 164025, 1399489, 6765201, 23532201, 66015625, 159138225, 342731169, 676572121, 1246160601, 2169230625, 3603000625, 5752160649, 8877596841, 13305853201, 19439330625, 27767223225, 38877191929, 53467775361

Offset: 0

Mohammad K. Azarian

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number

Cf. A002411, A014800, A015223, A015224.

A014799:=List(Filtered(List([0..10^2], n->n^2*(n+1)/2),IsOddInt),i->i^2); # Muniru A Asiru, Oct 09 2017

[(1+2*n)^2*(1+4*n)^4: n in [0..30]]; // Vincenzo Librandi, Apr 18 2012

seq( (1+2*n)^2*(1+4*n)^4, n=0..30); # Muniru A Asiru, Oct 09 2017

Table[(1+2*n)^2*(1+4*n)^4,{n,0,20}] (* Vincenzo Librandi, Apr 18 2012 *)
Range[0,20]//(1+2*#)^2*(1+4*#)^4& (* Waldemar Puszkarz, Oct 11 2017 *)

From Colin Barker, Apr 17 2012: (Start)
a(n) = (1 + 2*n)^2*(1 + 4*n)^4.
G.f.: (1 + 5618*x + 124671*x^2 + 369404*x^3 + 216463*x^4 + 21042*x^5 + 81*x^6) /(1-x)^7. (End)
a(n) = A015223(n)^2. - R. J. Mathar, Jul 30 2016

A014800 Squares of even pentagonal pyramidal numbers.

Original entry on oeis.org

0, 36, 324, 1600, 15876, 38416, 82944, 302500, 527076, 876096, 2160900, 3240000, 4734976, 9474084, 13032100, 17640000, 30980356, 40297104, 51840000, 83283876, 104162436, 129231424, 194602500, 236421376, 285474816, 409252900, 486202500

Offset: 0

Mohammad K. Azarian

Vincenzo Librandi, Table of n, a(n) for n = 0..1500
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-15,15,0,20,-20,0,-15,15,0,6,-6,0,-1,1).

Cf. A002411, A015224, A015223, A014799.

Select[CoefficientList[Series[(x(2x+1))/(x-1)^4,{x,0,50}],x],EvenQ]^2 (* Harvey P. Dale, Feb 27 2012 *)

G.f.: -4*x*(x^16 +143*x^15 +481*x^14 +1394*x^13 +9661*x^12 +10814*x^11 +15996*x^10 +45222*x^9 +25248*x^8 +23414*x^7 +33610*x^6 +9218*x^5 +5203*x^4 +3515*x^3 +319*x^2 +72*x +9)/((x -1)^7*(x^2 +x +1)^6). [Colin Barker, Nov 16 2012]

a(n) = A015224(n)^2. - R. J. Mathar, Jul 30 2016

A015223 Odd pentagonal pyramidal numbers.

Original entry on oeis.org

1, 75, 405, 1183, 2601, 4851, 8125, 12615, 18513, 26011, 35301, 46575, 60025, 75843, 94221, 115351, 139425, 166635, 197173, 231231, 269001, 310675, 356445, 406503, 461041, 520251, 584325, 653455, 727833, 807651, 893101, 984375

Offset: 0

Mohammad K. Azarian

Also first bisection of A139757. - Bruno Berselli, Feb 13 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002411, A014799, A015224, A014800.

[(2*n+1)*(4*n+1)^2: n in [0..50]]; // G. C. Greubel, Nov 04 2017

Table[((n+1)^3+(n+1)^2)/2,{n,0,200,4}] (* Vladimir Joseph Stephan Orlovsky, May 21 2011 *)
CoefficientList[Series[(1 + 71 x + 111 x^2 + 9 x^3)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 15 2013 *)

a(n)=(2*n+1)*(4*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1 + 71*x + 111*x^2 + 9*x^3)/(1-x)^4. - Colin Barker, Feb 13 2012
a(n) = (2n+1)*(4n+1)^2 = A130656(4n+1). - Bruno Berselli, Feb 13 2012
From Ant King, Oct 23 2012: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 192.
Sum_{n>=0} 1/a(n) = (8*C - 2*Pi + Pi^2 - 4*log(2))/8, where C is Catalan's constant (A006752). (End)
E.g.f.: (1 + 74*x + 128*x^2 + 32*x^3)*exp(x). - G. C. Greubel, Nov 04 2017

More terms from Erich Friedman

A015224 Even pentagonal pyramidal numbers.

Original entry on oeis.org

0, 6, 18, 40, 126, 196, 288, 550, 726, 936, 1470, 1800, 2176, 3078, 3610, 4200, 5566, 6348, 7200, 9126, 10206, 11368, 13950, 15376, 16896, 20230, 22050, 23976, 28158, 30420, 32800, 37926, 40678, 43560, 49726, 53016, 56448, 63750, 67626, 71656

Offset: 0

Mohammad K. Azarian

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (1, 0, 3, -3, 0, -3, 3, 0, 1, -1).

Cf. A002411, A014800, A015223, A014799, A006752.

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x*(3+6*x+11*x^2+34*x^3+17*x^4+13*x^5+11*x^6+x^7)/((1-x)^4*(1+x +x^2)^3))); // G. C. Greubel, Aug 24 2018

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{0,6,18,40,126,196,288,550, 726,936},40] (* Ant King, Oct 19 2012 *)

x='x+O('x^30); concat([0], Vec(2*x*(3+6*x+11*x^2+34*x^3 +17*x^4 +13*x^5+11*x^6+x^7)/((1-x)^4*(1+x +x^2)^3))) \\ G. C. Greubel, Aug 24 2018

From Ant King, Oct 24 2012: (Start)
a(n) = a(n-1) +3*a(n-3) -3*a(n-4) -3*a(n-6) +3*a(n-7) +a(n-9) -a(n-10).
a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) +192.
Sum_{n>=0} 1/a(n) = log(2)/2 + Pi/4 + 5*Pi^2/24 - 2 - C = 0.27217..., where C is Catalan's constant (A006752).
G.f.: 2*x*(3+6*x+11*x^2+34*x^3+17*x^4+13*x^5+11*x^6+x^7) / ((1-x)^4*(1+x +x^2)^3). (End)
a(n) = A002411(A004772(n+1)). - Bruno Berselli, Oct 24 2012

More terms from Patrick De Geest, Jul 14 1999

A079904 Triangle read by rows: T(n, k) = n*k, 0 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 2, 4, 0, 3, 6, 9, 0, 4, 8, 12, 16, 0, 5, 10, 15, 20, 25, 0, 6, 12, 18, 24, 30, 36, 0, 7, 14, 21, 28, 35, 42, 49, 0, 8, 16, 24, 32, 40, 48, 56, 64, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121

Offset: 0

Reinhard Zumkeller, Feb 21 2003

See the comment in A025581 on a problem posed by François Viète (Vieta) 1593, where this triangle is related to A025581 and A257238. - Wolfdieter Lang, May 12 2015

			Triangle T(n, k) begins:
  n\k 0  1  2  3  4  5  6  7  8  9  10 ...
  0:  0
  1:  0  1
  2:  0  2  4
  3:  0  3  6  9
  4:  0  4  8 12 16
  5:  0  5 10 15 20 25
  6:  0  6 12 18 24 30 36
  7:  0  7 14 21 28 35 42 49
  8:  0  8 16 24 32 40 48 56 64
  9:  0  9 18 27 36 45 54 63 72 81
  10: 0 10 20 30 40 50 60 70 80 90 100
  ... - _Wolfdieter Lang_, May 12 2015

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Cf. A005843, A008585, A008586, A005563, A002378, A000290.
Cf. A075362 (without column k=0), A002411 (row sums), A001105 (central terms).
Cf. A257238, A025581.

seq(seq(n*k, k=0..n), n=0..10); # Robert Israel, May 12 2015

Array[Range[0, #^2, #] &, 15, 0] (* Paolo Xausa, Mar 27 2025 *)

row(n) = vector(n+1, i, (i-1)*n); \\ Amiram Eldar, May 12 2025

T(n, k) = n*k, 0 <= k <= n.
T(n, k) = if k = 0 then 0 else T(n, k-1) + n.
T(n, 0) = 1. T(n, 1) = n for n > 0.
T(n, 2) = A005843(n) for n > 1.
T(n, 3) = A008585(n) for n > 2.
T(n, 4) = A008586(n) for n > 3.
T(n, n-2) = A005563(n-1) for n > 1.
T(n, n-1) = A002378(n-1) for n > 0.
T(n, n) = A000290(n).
T(n, k) = (A257238(n, k) - A025581(n, k)^3) / (3*A025581(n, k)). See the Viète comment above. - Wolfdieter Lang, May 12 2015
From Robert Israel, May 12 2015: (Start)
G.f. as triangle: (1 + x*y - 2*x^2*y)*x*y/((1 - x)^2*(1 - x*y)^3).
G.f. as sequence: -(Sum_{n>=0} (n^2 - n)*x^(n*(n + 1)/2)) / (1 - x) + (Sum_{n>=1} x^(n*(n + 1)/2)) * x/(1 - x)^2. These sums are related to Jacobi Theta functions. (End)
T(n, k) = gcd(n, k) * lcm(n, k). - Peter Luschny, Mar 26 2025

cp's OEIS Frontend

A261720 Array of pyramidal (3-dimensional figurate numbers) read by antidiagonals.

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

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Magma

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A267370 Partial sums of A140091.

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A285522 Array read by antidiagonals: T(m,n) = number of circulant digraphs up to Cayley isomorphism on n vertices with edges colored according to step value using a maximum of m-1 colors.

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A319184 Numbers that are sums of consecutive pentagonal numbers.

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A014799 Squares of odd pentagonal pyramidal numbers.

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A014800 Squares of even pentagonal pyramidal numbers.

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