login
A137680
Triangle read by rows, T(n,k) = T(n-1, k-1) - T(n-k, k-1); with leftmost term in each row = sum of all previous terms.
3
1, 1, 1, 3, 0, 1, 7, 2, 0, 1, 17, 4, 1, 0, 1, 40, 10, 4, 1, 0, 1, 96, 23, 8, 3, 1, 0, 1, 228, 56, 19, 8, 3, 1, 0, 1, 544, 132, 46, 18, 7, 3, 1, 0, 1, 1296, 316, 109, 42, 18, 7, 3, 1, 0, 1, 3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1, 7361, 1793, 620, 241, 98, 41, 17, 7, 3, 1, 0, 1, 17544, 4272, 1477, 574, 233, 97, 40, 17, 7, 3, 1, 0, 1
OFFSET
1,4
COMMENTS
A variation of the same sequence = column 2 of the triangle: (1, 0, 2, 4, 10, 23, 56, 132, ...) = first difference row of column 1. Left border of the triangle = A137682.
Left column starting (1, 3, ...) = INVERT transform of A160096. - Gary W. Adamson, May 01 2009
EXAMPLE
First few rows of the triangle:
1;
1, 1;
3, 0, 1;
7, 2, 0, 1;
17, 4, 1, 0, 1;
40, 10, 4, 1, 0, 1;
96, 23, 8, 3, 1, 0, 1;
228, 56, 19, 8, 3, 1, 0, 1;
544, 132, 46, 18, 7, 3, 1, 0, 1;
1296, 316, 109, 42, 18, 7, 3, 1, 0, 1;
3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1;
...
MAPLE
A137680 := proc(n, k)
if k < 1 or k > n then
0 ;
elif n = 1 then
1;
elif k = 1 then
add(add(procname(r, j), j=1..r), r=1..n-1) ;
else
procname(n-1, k-1)-procname(n-k, k-1) ;
end if;
end proc: # R. J. Mathar, Aug 12 2012
MATHEMATICA
T[n_, k_] := T[n, k] = Which[k < 1 || k > n, 0, n == 1, 1, k == 1, Sum[T[r, j], {r, 1, n-1}, {j, 1, r}], True, T[n-1, k-1] - T[n-k, k-1]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 02 2024, after R. J. Mathar *)
CROSSREFS
Cf. A137681 (row sums), A137682.
Cf. A160096. - Gary W. Adamson, May 01 2009
Sequence in context: A297786 A214407 A298668 * A248722 A201663 A199606
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Feb 05 2008
STATUS
approved