A116924 Triangle T(n,k) = B(k,n) - B(k-1,n) where B(n,m) = Sum_{i=0..n} binomial(m,i) (3*i+1).

1, 1, 4, 1, 8, 7, 1, 12, 21, 10, 1, 16, 42, 40, 13, 1, 20, 70, 100, 65, 16, 1, 24, 105, 200, 195, 96, 19, 1, 28, 147, 350, 455, 336, 133, 22, 1, 32, 196, 560, 910, 896, 532, 176, 25, 1, 36, 252, 840, 1638, 2016, 1596, 792, 225, 28, 1, 40, 315, 1200

Offset: 0

Author

Gary W. Adamson, Feb 26 2006

Keywords

• nonn
• tabl
• easy

Comments

The auxiliary array B(n,m) contains the binomial transform of the finite sequence of the first n+1 terms of 1, 4, 7, ... = A016777(.) in row n.

First few rows of the array B(n,m) are from row n=0 on:

1, 1, 1, 1, 1, ... (binomial transform of 1,0,0,0,...)

1, 5, 9, 13, 17, ... (binomial transform of 1,4,0,0,...)

1, 5, 16, 34, 59, ... (binomial transform of 1,4,7,0,...)

1, 5, 16, 44, 99, ... (binomial transform of 1,4,7,10,0,0,...)

The n-th row of triangle T contains the first differences of the n-th column of B.

Examples

			First few rows of the triangle T(n,k):
  1;
  1,  4;
  1,  8,   7;
  1, 12,  21,  10;
  1, 16,  42,  40,  13;
  1, 20,  70, 100,  65, 16;
  1, 24, 105, 200, 195, 96, 19;
  ...
		

Crossrefs

Cf. A016777, A053220.

Programs

Maple

B := proc(n,m) add(binomial(m,i)*(3*i+1),i=0..n) ; end proc:
A116924 := proc(n,k) if k = 0 then 1; else B(k,n)-B(k-1,n) ; end if; end proc:
seq(seq(A116924(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Mar 27 2010

Formula

Sum_{k=0..n} T(n,k) = A053220(n+1).

Extensions

Replaced definition with formula, corrected from 5th row on by R. J. Mathar, Mar 27 2010