A158823 Triangle read by rows: matrix product A004736 * A158821.

1, 3, 1, 6, 2, 2, 10, 3, 4, 3, 15, 4, 6, 6, 4, 21, 5, 8, 9, 8, 5, 28, 6, 10, 12, 12, 10, 6, 36, 7, 12, 15, 16, 15, 12, 7, 45, 8, 14, 18, 20, 20, 18, 14, 8, 55, 9, 16, 21, 24, 25, 24, 21, 16, 9, 66, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 78, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11

Offset: 1

Author

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

Keywords

• nonn
• tabl
• easy

Examples

			First few rows of the triangle =
   1;
   3,  1;
   6,  2,  2;
  10,  3,  4,  3;
  15,  4,  6,  6,  4;
  21,  5,  8,  9,  8,  5;
  28,  6, 10, 12, 12, 10,  6;
  36,  7, 12, 15, 16, 15, 12,  7;
  45,  8, 14, 18, 20, 20, 18, 14,  8;
  55,  9, 16, 21, 24, 25, 24, 21, 16,  9;
  66, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10;
  78, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11;
  91, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12;
		

Links

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

Crossrefs

Cf. A000292 (row sums), A003991, A004736, A158821.

Programs

Magma

[k eq 1 select Binomial(n+1, 2) else (n-k+1)*(k-1): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 01 2021

Maple

A158823 := proc(n,m) add( A004736(n,k)*A158821(k-1,m-1),k=1..n) ; end: seq(seq(A158823(n,m),m=1..n),n=1..8) ; # R. J. Mathar, Oct 22 2009

Mathematica

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

Sage

flatten([[binomial(n+1, 2) if k==1 else (n-k+1)*(k-1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 01 2021

Formula

Sum_{k=1..n} T(n, k) = A000292(n).

T(n, k) = Sum_{j=k..n} A004736(n, j)*A158821(j-1, k-1).

From R. J. Mathar, Mar 03 2011: (Start)

T(n, k) = (n-k+1)*(k-1), k>1.

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

Extensions

Corrected A-number in a formula - R. J. Mathar, Oct 30 2009