A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)*(n-k+1)*(n-k+2)/2, read by rows.

1, 4, 1, 10, 3, 2, 20, 6, 6, 3, 35, 10, 12, 9, 4, 56, 15, 20, 18, 12, 5, 84, 21, 30, 30, 24, 15, 6, 120, 28, 42, 45, 40, 30, 18, 7, 165, 36, 56, 63, 60, 50, 36, 21, 8, 220, 45, 72, 84, 84, 75, 60, 42, 24, 9, 286, 55, 90, 108, 112, 105, 90, 70, 48, 27, 10, 364, 66, 110, 135, 144, 140, 126, 105, 80, 54, 30, 11

Offset: 1

Author

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

Keywords

• nonn
• tabl
• easy

Comments

The triangle can also be defined by multiplying the triangles A(n,k)=1 and A158823(n,k), that is, this here are the partial column sums of A158823.

Examples

			First few rows of the triangle are:
    1;
    4,  1;
   10,  3,   2;
   20,  6,   6,   3;
   35, 10,  12,   9,   4;
   56, 15,  20,  18,  12,   5;
   84, 21,  30,  30,  24,  15,   6;
  120, 28,  42,  45,  40,  30,  18,   7;
  165, 36,  56,  63,  60,  50,  36,  21,   8;
  220, 45,  72,  84,  84,  75,  60,  42,  24,  9;
  286, 55,  90, 108, 112, 105,  90,  70,  48, 27, 10;
  364, 66, 110, 135, 144, 140, 126, 105,  80, 54, 30, 11;
  455, 78, 132, 165, 180, 180, 168, 147, 120, 90, 60, 33, 12;
  ...
		

Links

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

Crossrefs

Cf. A062707, A104633, A158823.

Row sums: A000332.

Programs

Magma

A158824:= func< n,k | k eq 1 select Binomial(n+2,3) else (k-1)*Binomial(n-k+2,2) >; [A158824(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021

Mathematica

T[n_, k_]:= If[k==1, Binomial[n+2, 3], (k-1)*Binomial[n-k+2, 2]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

Sage

def A158824(n,k): return binomial(n+2,3) if k==1 else (k-1)*binomial(n-k+2,2)
flatten([[A158824(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021

Formula

T(n,k) = binomial(n+2,3) if k = 1 otherwise (k-1)*binomial(n-k+2, 2).

Sum_{k=1..n} T(n, k) = binomial(n+3, 4) = A000332(n+3). - G. C. Greubel, Apr 01 2021