A158822 Triangle read by rows, matrix triple product A000012 * A145677 * A000012.

1, 3, 1, 6, 3, 2, 10, 6, 5, 3, 15, 10, 9, 7, 4, 21, 15, 14, 12, 9, 5, 28, 21, 20, 18, 15, 11, 6, 36, 28, 27, 25, 22, 18, 13, 7, 45, 36, 35, 33, 30, 26, 21, 15, 8, 55, 45, 44, 42, 39, 35, 30, 24, 17, 9, 66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10

Offset

0,2

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

Triangle read by rows, A000012 * A145677 * A000012; where A000012 = an infinite lower triangular matrix: (1; 1,1; 1,1,1; ...), with all 1's.

From G. C. Greubel, Dec 26 2021: (Start)

T(n, k) = (n+1-k)*(n+k)/2 with T(n, 0) = binomial(n+2, 2).

Sum_{k=0..n} T(n, k) = (1/3)*(n+1)*(n^2 + 2*n + 3) = A006527(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = binomial(n+2, 2) + A034828(n+1).

T(n, 1) = A000217(n).

T(n, 2) = A000096(n-1).

T(n, 3) = A055998(n-2).

T(2*n, n) = A134479(n). (End)

Example

First few rows of the triangle =
   1;
   3,  1;
   6,  3,  2;
  10,  6,  5,  3;
  15, 10,  9,  7,  4;
  21, 15, 14, 12,  9,  5;
  28, 21, 10, 18, 15, 11,  6;
  36, 28, 27, 25, 22, 18, 13,  7;
  45, 36, 35, 33, 30, 26, 21, 15,  8;
  55, 45, 44, 42, 39, 35, 30, 24, 17,  9;
  66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10;
  78, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11;
  91, 78, 77, 75, 72, 68, 63, 57, 50, 42, 33, 23, 12;
  ...

Mathematica

T[n_, k_]:= If[k==0, Binomial[n+2, 2], (n+1-k)*(n+k)/2];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 26 2021 *)

Prog

(Sage)
def A158822(n, k):
    if (k==0): return binomial(n+2, 2)
    else: return (n-k+1)*(n+k)/2
flatten([[A158822(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 26 2021

Crossrefs

Cf. A000012, A000096, A000217, A006527, A034828, A055998, A145677, A134479.

Sequence in context: A107884 A185628 A321194 * A353924 A226132 A121443

Adjacent sequences: A158819 A158820 A158821 * A158823 A158824 A158825

Keyword

nonn,tabl

Author

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

Extensions

Definition corrected by Michael Somos, Nov 05 2011

Status

approved