A006000 -id:A006000 - cp's OEIS frontend

A133252 Partial sums of A006000.

1, 5, 17, 45, 100, 196, 350, 582, 915, 1375, 1991, 2795, 3822, 5110, 6700, 8636, 10965, 13737, 17005, 20825, 25256, 30360, 36202, 42850, 50375, 58851, 68355, 78967, 90770, 103850, 118296, 134200, 151657, 170765, 191625, 214341, 239020, 265772

Offset: 0

Jonathan Vos Post, Dec 19 2007

Prime for a(1) = 5, a(2) = 17, then never again?

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

Cf. A006000.

LinearRecurrence[{5,-10,10,-5,1},{1,5,17,45,100},40] (* Harvey P. Dale, Sep 15 2022 *)

a(n) = Sum_{i=0..n} A006000(i).
a(n) = Sum_{i=0..n} (i+1)*(i^2+i+2)/2.
a(n) = ((n^4+2*n^3+n^2)/4+(2*n^3+3*n^2+n)/3+(3*n^2+3*n)/2+2*n)/2+1.
G.f.: -(2*x^2 + 1) / (x-1)^5. - Colin Barker, Apr 28 2013
a(n) = (n+1)*(n+2)*(3*n^2+5*n+12)/24. - Alois P. Heinz, Apr 28 2013
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, Apr 21 2024

A285362 Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 4, 2, 15, 12, 3, 60, 58, 28, 4, 262, 273, 185, 55, 5, 1243, 1329, 1094, 495, 96, 6, 6358, 6839, 6293, 3757, 1148, 154, 7, 34835, 37423, 36619, 26421, 11122, 2380, 232, 8, 203307, 217606, 219931, 180482, 96454, 28975, 4518, 333, 9, 1257913, 1340597, 1376929, 1230737, 787959, 308127, 67898, 7995, 460, 10

Offset: 1

Alois P. Heinz, Apr 17 2017

			T(3,2) = 12 because the sum of the entries in the second blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+3+2+5+2 = 12.
Triangle T(n,k) begins:
      1;
      4,     2;
     15,    12,     3;
     60,    58,    28,     4;
    262,   273,   185,    55,     5;
   1243,  1329,  1094,   495,    96,    6;
   6358,  6839,  6293,  3757,  1148,  154,   7;
  34835, 37423, 36619, 26421, 11122, 2380, 232, 8;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Partition of a set

Columns k=1-10 give: A285363, A285364, A285365, A285366, A285367, A285368, A285369, A285370, A285371, A285372.
Row sums give A000110(n) * A000217(n) = A105488(n+3).
Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).
T(2n+1,n+1) gives A285410.
Cf. A270236, A285439, A285595.

T:= proc(h) option remember; local b; b:=
      proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> p
        +[0, (h-n+1)*p[1]*x^j])(b(n-1, max(m, j))), j=1..m+1))
      end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, 0)[2])
    end:
seq(T(n), n=1..12);

T[h_] := T[h] = Module[{b}, b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[# + {0, (h - n + 1)*#[[1]]*x^j}&[b[n - 1, Max[m, j]]], {j, 1, m + 1}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, 0][[2]]]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 5, 9, 10, 0, 1, 6, 12, 16, 15, 0, 1, 7, 15, 22, 25, 21, 0, 1, 8, 18, 28, 35, 36, 28, 0, 1, 9, 21, 34, 45, 51, 49, 36, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 0, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66

Offset: 0

Omar E. Pol, Apr 27 2008

A general formula for polygonal numbers is P(n,k) = (n-2)(k-1)k/2 + k, where P(n,k) is the k-th n-gonal number. - Omar E. Pol, Dec 21 2008

			The square array of polygonal numbers begins:
========================================================
Triangulars .. A000217: 0, 1,  3,  6, 10,  15,  21,  28,
Squares ...... A000290: 0, 1,  4,  9, 16,  25,  36,  49,
Pentagonals .. A000326: 0, 1,  5, 12, 22,  35,  51,  70,
Hexagonals ... A000384: 0, 1,  6, 15, 28,  45,  66,  91,
Heptagonals .. A000566: 0, 1,  7, 18, 34,  55,  81, 112,
Octagonals ... A000567: 0, 1,  8, 21, 40,  65,  96, 133,
9-gonals ..... A001106: 0, 1,  9, 24, 46,  75, 111, 154,
10-gonals .... A001107: 0, 1, 10, 27, 52,  85, 126, 175,
11-gonals .... A051682: 0, 1, 11, 30, 58,  95, 141, 196,
12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217,
And so on ..............................................
========================================================

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.

Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
Cf. A000007, A000012, A000027, A002411, A006000, A006003, A006522.
Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.
Cf. A139608, A139609, A139610, A139611, A139612, A139613, A139614.
Cf. A139615, A139616, A057145, A086271, A139600, A139617, A139618.
Cf. A139619, A139620.

T:= func< n,k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n,k | T(n-k, k) >;
[A139601(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def T(n,k): return k*((n+1)*(k-1)+2)/2
def A139601(n,k): return T(n-k, k)
flatten([[A139601(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = A086270(n,k), k>0. - R. J. Mathar, Aug 06 2008
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009
From G. C. Greubel, Jul 12 2024: (Start)
t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
t(2*n, n) = A006003(n).
t(2*n+1, n) = A002411(n).
t(2*n-1, n) = A006000(n-1).
Sum_{k=0..n} t(n, k) = A006522(n+2).
Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).
Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)

A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 9, 16, 25, 1, 5, 12, 22, 35, 51, 1, 6, 15, 28, 45, 66, 91, 1, 7, 18, 34, 55, 81, 112, 148, 1, 8, 21, 40, 65, 96, 133, 176, 225, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451

Offset: 0

Paul Barry, Mar 21 2003

			The array starts
  1  1  3 10 ...
  1  2  6 16 ...
  1  3  9 22 ...
  1  4 12 28 ...
The triangle starts
  1;
  1,  1;
  1,  2,  3;
  1,  3,  6, 10;
  1,  4,  9, 16, 25;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Polygonal Number

Rows include A060354, A064808, A006000, A006003, A002411.
Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.
Antidiagonals are composed of n-gonal numbers.

Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # G. C. Greubel, Aug 14 2019

[[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // G. C. Greubel, Oct 13 2018

Table[PolygonalNumber[n,i],{n,0,10},{i,n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* Harvey P. Dale, Aug 27 2016 *)

tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

[[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # G. C. Greubel, Aug 14 2019

Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.

T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - Michel Marcus, Jun 22 2015

A086271 Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 9, 10, 1, 6, 12, 16, 15, 1, 7, 15, 22, 25, 21, 1, 8, 18, 28, 35, 36, 28, 1, 9, 21, 34, 45, 51, 49, 36, 1, 10, 24, 40, 55, 66, 70, 64, 45, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66, 1, 13, 33, 58, 85, 111, 133, 148, 153, 145, 121, 78

Offset: 1

Clark Kimberling, Jul 14 2003

The transpose of the array in A086270; diagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.

			Columns 1,2,3 are the triangular, square and pentagonal numbers.
Northwest corner:
       k=1 k=2 k=3 k=4 k=5
  n=1:   1   1   1   1   1 ...
  n=2:   3   4   5   6   7 ...
  n=3:   6   9  12  15  18 ...
  n=4:  10  16  22  28  34 ...
  n=5:  15  25  35  45  55 ...
  ...

G. C. Greubel, Table of n, a(n) for the first 50 diagonals, flattened
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A006522, A086270, A086272, A086273.

Main diagonal gives A006000(n-1).

T[n_, k_] := PolygonalNumber[k+2, n]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 04 2016 *)

T(n, k) = k*C(n,2) + n.
From Stefano Spezia, Sep 02 2022: (Start)
G.f.: x*y*(1 - y + x*y)/((1 - x)^3*(1 - y)^2).
G.f. of n-th row: n*(1 + n - 2*y)*y/(2*(1 - y)^2). (End)

A285439 Sum T(n,k) of the entries in the k-th cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 4, 2, 21, 12, 3, 132, 76, 28, 4, 960, 545, 235, 55, 5, 7920, 4422, 2064, 612, 96, 6, 73080, 40194, 19607, 6692, 1386, 154, 7, 745920, 405072, 202792, 75944, 18736, 2816, 232, 8, 8346240, 4484808, 2280834, 911637, 254061, 46422, 5256, 333, 9

Offset: 1

Alois P. Heinz, Apr 19 2017

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

			T(3,1) = 21 because the sum of the entries in the first cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 6+6+3+4+1+1 = 21.
Triangle T(n,k) begins:
       1;
       4,      2;
      21,     12,      3;
     132,     76,     28,     4;
     960,    545,    235,    55,     5;
    7920,   4422,   2064,   612,    96,    6;
   73080,  40194,  19607,  6692,  1386,  154,   7;
  745920, 405072, 202792, 75944, 18736, 2816, 232, 8;
  ...

Alois P. Heinz, Rows n = 1..23, flattened
Wikipedia, Permutation

Columns k=1-2 give: A284816, A285489.
Row sums give A000142 * A000217 = A180119.
Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).
Cf. A000290, A002775, A185105, A285362, A285382, A285793.

T:= proc(h) option remember; local b; b:=
      proc(n, l) option remember; `if`(n=0, [mul((i-1)!, i=l), 0],
        (p-> p+[0, (h-n+1)*p[1]*x^(nops(l)+1)])(b(n-1, [l[], 1]))+
         add((p-> p+[0, (h-n+1)*p[1]*x^j])(
         b(n-1, subsop(j=l[j]+1, l))), j=1..nops(l)))
      end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, [])[2])
    end:
seq(T(n), n=1..10);

T[h_] := T[h] = Module[{b}, b[n_, l_] := b[n, l] = If[n == 0, {Product[(i - 1)!, {i, l}], 0}, # + {0, (h - n + 1)*#[[1]]*x^(Length[l] + 1)}&[b[n - 1, Append[l, 1]]] + Sum[# + {0, (h-n+1)*#[[1]]*x^j}&[b[n - 1, ReplacePart[ l, j -> l[[j]] + 1]]], {j, 1, Length[l]}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, {}][[2]]]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)

Sum_{k=1..n} k * T(n,k) = n^2 * n! = A002775(n).

A128227 Right border (1,1,1,...) added to A002260.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1

Offset: 0

Gary W. Adamson, Feb 19 2007

Row sums = A000124: (1, 2, 4, 7, 11, 16, ...). n* each term of the triangle gives A128228, having row sums A006000: (1, 4, 12, 28, 55, ...).
Eigensequence of the triangle = A005425: (1, 2, 5, 14, 43, ...). - Gary W. Adamson, Aug 27 2010
From Franck Maminirina Ramaharo, Aug 25 2018: (Start)
T(n,k) is the number of binary words of length n having k letters 1 such that no 1's lie between any pair of 0's.
Let n lines with equations y = (i - 1)*x - (i - 1)^2, i = 1..n, be drawn in the Cartesian plane. For each line, call the half plane containing the point (-1,1) the upper half plane and the other half the lower half-plane. Then T(n,k) is the number of regions that are the intersections of k upper half-planes and n-k lower half-planes. Here, T(0,0) = 1 corresponds to the plane itself. A region obtained from this arrangement of lines can be associated with a length n binary word such that the i-th letter indicates whether the region is located at the i-th upper half-plane (letter 1) or at the lower half-plane (letter 0).
(End)

			First few rows of the triangle are:
1;
1, 1;
1, 2, 1;
1, 2, 3, 1;
1, 2, 3, 4, 1;
1, 2, 3, 4, 5, 1;
1, 2, 3, 4, 5, 6, 1;
1, 2, 3, 4, 5, 6, 7, 1;
1, 2, 3, 4, 5, 6, 7, 8, 1;
...
From _Franck Maminirina Ramaharo_, Aug 25 2018: (Start)
For n = 5, the binary words are
(k = 0) 00000;
(k = 1) 10000, 00001;
(k = 2) 11000, 10001, 00011;
(k = 3) 11100, 11001, 10011, 00111;
(k = 4) 11110, 11101, 11011, 10111, 01111;
(k = 5) 11111.
(End)

A. Bogolmony, Number of Regions N Lines Divide Plane.
Eric Weisstein's World of Mathematics, Plane Division by Lines.
J. E. Wetzel, On the division of the plane by lines, The American Mathematical Monthly Vol. 85 (1978), 647-656.

Cf. A002260, A128228, A000124, A006000, A318274.

Cf. A005425. - Gary W. Adamson, Aug 27 2010

(* first n rows of the triangle *)
a128227[n_] := Table[If[r==q, 1, q], {r, 1, n}, {q, 1, r}]
Flatten[a128227[13]] (* data *)
TableForm[a128227[5]] (* triangle *)
(* Hartmut F. W. Hoft, Jun 10 2017 *)

T(n, k) := if n = k then 1 else k + 1$
for n:0 thru 10 do print(makelist(T(n, k), k, 0, n)); /* Franck Maminirina Ramaharo, Aug 25 2018 */

def T(n, k): return 1 if n==k else k
for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 10 2017

from math import comb, isqrt
def A128227(n): return n-comb(r:=(m:=isqrt(k:=n+1<<1))+(k>m*(m+1))+1,2)+(2 if k==m*(m+1) else r) # Chai Wah Wu, Nov 09 2024

"1" added to each row of "start counting again": (1; 1,2; 1,2,3,...) such that a(1) = 1, giving: (1; 1,1; 1,2,1;...).
T(n,k) = k if 1<=kHartmut F. W. Hoft, Jun 10 2017
From Franck Maminirina Ramaharo, Aug 25 2018: (Start)
The n-th row are the coefficients in the expansion of ((x^2 + (n - 2)*x - n)*x^n + 1)/(x - 1)^2.
G.f. for column k: ((k*x + 1)*x^k)/(1 - x). (End)

A256859 a(n) = n(n + 1)(n + 2)*(n^2 - n + 4)/24.

Original entry on oeis.org

1, 6, 25, 80, 210, 476, 966, 1800, 3135, 5170, 8151, 12376, 18200, 26040, 36380, 49776, 66861, 88350, 115045, 147840, 187726, 235796, 293250, 361400, 441675, 535626, 644931, 771400, 916980, 1083760, 1273976, 1490016, 1734425, 2009910, 2319345, 2665776, 3052426

Offset: 1

Luciano Ancora, Apr 14 2015

This is the case k = n of b(n,k) = n*(n+1)*(n+2)*(k*(n-1)+4)/24, where b(n,k) is the n-th hypersolid number in 4 dimensions generated from an arithmetical progression with the first term 1 and common difference k. Therefore, the sequence is the main diagonal of the Table 3 in Sardelis et al. paper (see Links field).

G. C. Greubel, Table of n, a(n) for n = 1..1000
D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070v1 [math.GM], 2008.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A261721.

Cf. similar sequences of the form binomial(n+k-2,k-1)+n*binomial(n+k-2,k): A006000 (k=2); A257055 (k=3); this sequence (k=4); A256860 (k=5); A256861 (k=6).

[n*(n + 1)*(n + 2)*(n^2 - n + 4)/24: n in [1..30]]; // G. C. Greubel, Nov 23 2017

Table[n (n + 1) (n + 2) (n^2 - n + 4)/24, {n, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,6,25,80,210,476},40] (* Harvey P. Dale, Mar 19 2022 *)

vector(40, n, n*(n+1)*(n+2)*(n^2-n+4)/24) \\ Bruno Berselli, Apr 15 2015

G.f.: x*(1 + 4*x^2)/(1 - x)^6.
a(n) = 4*A000389(n+2) + A000389(n+4). - Bruno Berselli, Apr 15 2015
E.g.f.: (24*x + 48*x^2 + 40*x^3 + 12*x^4 + x^5)*exp(x)/24. - G. C. Greubel, Nov 23 2017
a(n) = A261721(n,n-1). - Alois P. Heinz, Apr 15 2020

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 14, 14, 11, 5, 6, 16, 22, 24, 22, 16, 6, 7, 22, 32, 37, 37, 32, 22, 7, 8, 29, 44, 53, 56, 53, 44, 29, 8, 9, 37, 58, 72, 79, 79, 72, 58, 37, 9, 10, 46, 74, 94, 106, 110, 106, 94, 74, 46, 10, 11, 56, 92, 119

Offset: 0

Franck Maminirina Ramaharo, Mar 05 2018

Antidiagonal sums are given by 2*A055795.
Rows/columns n are binomial transform of {n, A152947(n+1), n, 0, 0, 0, ...}.
Some primes in the array are
n = 1: {2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, ...} = A055469, primes of the form k*(k + 1)/2 + 1;
n = 3: {3, 7, 37, 53, 479, 653, 1249, 1619, 2503, 3727, 4349, 5737, 7109, 8179, 9803, 11839, 12107, ...};
n = 4: {11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, ...} = A188382, primes of the form 8*(2*k - 1)^2 + 2*(2*k - 1) + 1.

			The array T(n,k) begins
0     1    2    3    4     5     6     7     8     9    10    11  ...
1     2    4    7   11    16    22    29    37    46    56    67  ...
2     4    8   14   22    32    44    58    74    92   112   134  ...
3     7   14   24   37    53    72    94   119   147   178   212  ...
4    11   22   37   56    79   106   137   172   211   254   301  ...
5    16   32   53   79   110   146   187   233   284   340   401  ...
6    22   44   72  106   146   192   244   302   366   436   512  ...
7    29   58   94  137   187   244   308   379   457   542   634  ...
8    37   74  119  172   233   302   379   464   557   658   767  ...
9    46   92  147  211   284   366   457   557   666   784   911  ...
10   56  112  178  254   340   436   542   658   784   920  1066  ...
11   67  134  212  301   401   512   634   767   911  1066  1232  ...
12   79  158  249  352   467   594   733   884  1047  1222  1409  ...
13   92  184  289  407   538   682   839  1009  1192  1388  1597  ...
14  106  212  332  466   614   776   952  1142  1346  1564  1796  ...
15  121  242  378  529   695   876  1072  1283  1509  1750  2006  ...
16  137  274  427  596   781   982  1199  1432  1681  1946  2227  ...
17  154  308  479  667   872  1094  1333  1589  1862  2152  2459  ...
18  172  344  534  742   968  1212  1474  1754  2052  2368  2702  ...
19  191  382  592  821  1069  1336  1622  1927  2251  2594  2956  ...
20  211  422  653  904  1175  1466  1777  2108  2459  2830  3221  ...
...
The inverse binomial transforms of the columns are
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
1     1    2    4    7    11    22    29    37    45    56    67  ...  A152947
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
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  ...
...

Miklós Bóna, Introduction to Enumerative Combinatorics, McGraw-Hill, 2007.
L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, 1974.
R. P. Stanley, Enumerative Combinatorics, second edition, Cambridge University Press, 2011.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.

Cf. A000124, A001477, A006000, A008815, A014206, A051601, A055469, A077028, A081436, A084849, A131074, A134394, A139600, A141387, A179000, A188377, A188382, A273465.

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

T[n_, k_] := n (Binomial[k, 2] + 1) + k (Binomial[n, 2] + 1);
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2018 *)

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

T(n, k) = n*(binomial(k,2) + 1) + k*(binomial(n,2) + 1);
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 12 2018

T(n,k) = T(k,n) = n*A152947(k+1) + k*A152947(n+1).
T(n,0) = A001477(n).
T(n,1) = A000124(n).
T(n,2) = A014206(n).
T(n,3) = A273465(3*n+2).
T(n,4) = A084849(n+1).
T(n,n) = A179000(n-1,n), n >= 1.
T(2*n,2*n) = 8*A081436(n-1), n >= 1.
T(2*n+1,2*n+1) = 2*A006000(2*n+1).
T(n,n+1) = A188377(n+3).
T(n,n+2) = A188377(n+2), n >= 1.
Sum_{k=0..n} T(k,n-k) = 2*(binomial(n, 4) + binomial(n, 2)).
G.f.: -((2*x*y - y - x)*(2*x*y - y - x + 1))/(((x - 1)*(y - 1))^3).
E.g.f.: (1/2)*(x + y)*(x*y + 2)*exp(x + y).

A128228 A128229 * A002260.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 8, 12, 4, 5, 10, 15, 20, 5, 6, 12, 18, 24, 30, 6, 7, 14, 21, 28, 35, 42, 7, 8, 16, 24, 32, 40, 48, 56, 8, 9, 18, 27, 36, 45, 54, 63, 72, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 10

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = A006000: (1, 4, 12, 28, 55, 96, 154,...).

			First few rows of the triangle are:
1;
2, 2;
3, 6, 3;
4, 8, 12, 4;
5, 10, 15, 20, 5;
6, 12, 18, 24, 30, 6;
7, 14, 21, 28, 35, 42, 7;
...

Cf. A128229, A002260, A006000, A128227.

(* first n rows of the triangle *)
a128228[n_] := Table[If[r==q, r, q r], {r, 1, n}, {q, 1, r}]
Flatten[a128228[10]] (* data *)
TableForm[a128228[7]] (* triangle *)
(* Hartmut F. W. Hoft, Jun 10 2017 *)

A128229 * A002260 as infinite lower triangular matrices.
Triangle, n * (each term of A128227).
T(n,k) = k*n if 1<=kHartmut F. W. Hoft, Jun 10 2017

cp's OEIS Frontend

A133252 Partial sums of A006000.

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A285362 Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k.

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A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

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GAP

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A086271 Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.

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A285439 Sum T(n,k) of the entries in the k-th cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A128227 Right border (1,1,1,...) added to A002260.

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A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

A256859 a(n) = n(n + 1)(n + 2)*(n^2 - n + 4)/24.

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.