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

0,3

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.

Links

Table of n, a(n) for n=0..58.

Formula

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

Example

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;
  ...

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

Crossrefs

Cf. A016777, A053220.

Sequence in context: A259930 A103553 A067439 * A327957 A335707 A232816

Adjacent sequences: A116921 A116922 A116923 * A116925 A116926 A116927

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Feb 26 2006

Extensions

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

Status

approved