A135853 A103516 * A007318 as an infinite lower triangular matrix.

1, 4, 2, 6, 6, 3, 8, 12, 12, 4, 10, 20, 30, 20, 5, 12, 30, 60, 60, 30, 6, 14, 42, 105, 140, 105, 42, 7, 16, 56, 168, 280, 280, 168, 56, 8, 18, 72, 252, 504, 630, 504, 252, 72, 9, 20, 90, 360, 840, 1260, 1260, 840, 360, 90, 10

Offset: 0

Author

Gary W. Adamson, Dec 01 2007

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle are:
   1;
   4,   2;
   6,   6,  3;
   8,  12,  12,   4;
  10,  20,  30,  20,   5;
  12,  30,  60,  60,  30,   6;
  14,  42, 105, 140, 105,  42,   7;
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows

Crossrefs

Cf. A007318, A103516.

Cf. A103517 (1st column), A135854 (row sums).

Cf. A135852 (= A007318 * A103516).

Programs

Mathematica

T[n_, k_]:= If[k==n, n+1, If[k==0, 2*(n+1), (k+1)*Binomial[n+1, k+1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//flatten (* G. C. Greubel, Dec 07 2016 *)

Sage

def A135853(n,k):
    if (n==0): return 1
    elif (k==0): return 2*(n+1)
    else: return (k+1)*binomial(n+1, k+1)
flatten([[A135853(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 06 2022

Formula

T(n, k) = (A103516 * A007318)(n, k).

Sum_{k=0..n} T(n, k) = A135854(n).

T(n, k) = (k+1)*binomial(n+1, k+1), with T(n, n) = n+1, T(n, 0) = 2*(n+1). - G. C. Greubel, Dec 07 2016

T(n, 0) = A103517(n). - G. C. Greubel, Feb 06 2022