OFFSET
0,5
COMMENTS
The right hand columns of triangle T(n, k) represent the Kn2p sums of the ‘Races with Ties’ triangle A035317. See A180662 for the definitions of these sums.
The row sums lead to A094687, the convolution of Fibonacci and Jacobsthal numbers, and the alternating row sums lead to A008346.
The backwards antidiagonal sums equal Kn21(n) = (-1)^n*A175722(n).
FORMULA
T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = F(n+2) - (1-(-1)^n)/2 = A052952(n), with F(n) = A000045(n), the Fibonacci numbers, n >= 0 and 0 <= k <= n.
T(n+p-1, n) = sum(A035317(n-k+p-1, n-2*k), k=0..floor(n/2)), n >= 0 and p >= 1.
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
Tsq(n, k) = sum(Tsq(n-1, i), i=0..k), n >= 1 and k >= 0, with Tsq(0, k) = A052952(k).
Tsq(n, k) = sum(A035317(n+k-i, k-2*i), i=0..floor(k/2)), n >= 0 and k >= 0.
The G.f. generates the terms in the n-th row of the square array Tsq(n, k).
G.f.: (-1)^(n)/((-1+x+x^2)*(x+1)*(x-1)^(n+1)), n >= 0.
EXAMPLE
The first few rows of triangle T(n, k), n >= 0 and 0 <= k <= n.
n/k 0 1 2 3 4 5 6 7
------------------------------------------------
0| 1
1| 1, 1
2| 1, 2, 3
3| 1, 3, 5, 4
4| 1, 4, 8, 9, 8
5| 1, 5, 12, 17, 17, 12
6| 1, 6, 17, 29, 34, 29, 21
7| 1, 7, 23, 46, 63, 63, 50, 33
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
n/k 0 1 2 3 4 5 6 7
------------------------------------------------
0| 1, 1, 3, 4, 8, 12, 21, 33
1| 1, 2, 5, 9, 17, 29, 50, 83
2| 1, 3, 8, 17, 34, 63, 113, 196
3| 1, 4, 12, 29, 63, 126, 239, 435
4| 1, 5, 17, 46, 109, 235, 474, 909
5| 1, 6, 23, 69, 178, 413, 887, 1796
6| 1, 7, 30, 99, 277, 690, 1577, 3373
7| 1, 8, 38, 137, 414, 1104, 2681, 6054
MAPLE
T:= proc(n, k) option remember: if k=0 then return(1) elif k=n then return(combinat[fibonacci](n+2) - (1-(-1)^n)/2) else procname(n-1, k-1)+procname(n-1, k) fi: end: seq(seq(T(n, k), k=0..n), n=0..10); # End first program.
KEYWORD
AUTHOR
Johannes W. Meijer, Oct 19 2013
STATUS
approved