A135852 A007318 * A103516 as a lower triangular matrix.

1, 3, 2, 8, 4, 3, 20, 6, 9, 4, 48, 8, 18, 16, 5, 112, 10, 30, 40, 25, 6, 256, 12, 45, 80, 75, 36, 7, 576, 14, 63, 140, 175, 126, 49, 8, 1280, 16, 84, 224, 350, 336, 196, 64, 9, 2816, 18, 108, 336, 630, 756, 588, 288, 81, 10

Offset: 0

Author

Gary W. Adamson, Dec 01 2007

Keywords

• nonn
• tabl

Comments

Binomial transform of triangle A103516.

Examples

			First few rows of the triangle are:
    1;
    3,  2;
    8,  4,  3;
   20,  6,  9,  4;
   48,  8, 18, 16,  5;
  112, 10, 30, 40, 25,  6;
  256, 12, 45, 80, 75, 36,  7;
  ...
		

Links

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

Crossrefs

Cf. A007318, A103516.

Cf. A001792 (1st column), A099035 (row sums).

Cf. A135853 (= A103516 * A007318).

Programs

Mathematica

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

Sage

def A135852(n,k):
    if (n==0): return 1
    elif (k==0): return (n+2)*2^(n-1)
    else: return (k+1)*binomial(n, k)
flatten([[A135852(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2022

Formula

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

T(n, 0) = A001792(n).

Sum_{k=0..n} T(n, k) = A099035(n+1).

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

Extensions

Offset changed to 0 by G. C. Greubel, Feb 07 2022