A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

0, 1, 4, 3, 7, 12, 6, 11, 17, 24, 10, 16, 23, 31, 40, 15, 22, 30, 39, 49, 60, 21, 29, 38, 48, 59, 71, 84, 28, 37, 47, 58, 70, 83, 97, 112, 36, 46, 57, 69, 82, 96, 111, 127, 144, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220

Offset: 0

Author

Reinhard Zumkeller, Jul 17 2014

Keywords

• nonn
• tabl

Examples

			First rows and their row sums (A245301):
   0                                                                  0;
   1,  4                                                              5;
   3,  7,  12                                                        22;
   6, 11,  17,  24                                                   58;
  10, 16,  23,  31,  40                                             120;
  15, 22,  30,  39,  49,  60                                        215;
  21, 29,  38,  48,  59,  71,  84                                   350;
  28, 37,  47,  58,  70,  83,  97, 112                              532;
  36, 46,  57,  69,  82,  96, 111, 127, 144                         768;
  45, 56,  68,  81,  95, 110, 126, 143, 161, 180                   1065;
  55, 67,  80,  94, 109, 125, 142, 160, 179, 199, 220              1430;
  66, 79,  93, 108, 124, 141, 159, 178, 198, 219, 241, 264         1870;
  78, 92, 107, 123, 140, 158, 177, 197, 218, 240, 263, 287, 312    2392.
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Crossrefs

Cf. A000217, A046092, A062725, A245301.

Programs

Haskell

a245300 n k = (n + k) * (n + k + 1) `div` 2 + k
a245300_row n = map (a245300 n) [0..n]
a245300_tabl = map a245300_row [0..]
a245300_list = concat a245300_tabl

Magma

[k + Binomial(n+k+1,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021

Mathematica

Table[k + Binomial[n+k+1,2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

Sage

flatten([[k + binomial(n+k+1,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 01 2021

Formula

T(n, 0) = A000217(n).

T(n, n) = A046092(n).

T(2*n, n) = A062725(n) (central terms).

Sum_{k=0..n} T(n, k) = A245301(n).

From G. C. Greubel, Apr 01 2021: (Start)

T(n, 1) = A000124(n+1) = A134869(n+1), n >= 1.

T(n, 2) = A152948(n+4), n >= 2.

T(n, 3) = A152950(n+4), n >= 3.

T(n, 4) = A145018(n+5), n >= 4.

T(n, 5) = A167499(n+4), n >= 5.

T(n, 6) = A166136(n+5), n >= 6.

T(n, 7) = A167487(n+6), n >= 7.

T(n, n-1) = A056220(n), n >= 1.

T(n, n-2) = A142463(n-1), n >= 2.

T(n, n-3) = A054000(n-1), n >= 3.

T(n, n-4) = A090288(n-3), n >= 4.

T(n, n-5) = A268581(n-4), n >= 5.

T(n, n-6) = A059993(n-4), n >= 6.

T(n, n-7) = (-1)*A147973(n), n >= 7.

T(n, n-8) = A139570(n-5), n >= 8.

T(n, n-9) = A271625(n-5), n >= 9.

T(n, n-10) = A222182(n-4), n >= 10.

T(2*n, n-1) = A081270(n-1), n >= 1.

T(2*n, n+1) = A117625(n+1), n >= 1. (End)