OFFSET
0,6
COMMENTS
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) * F(n-1) = A226205(n+1) with F(n) = A000045(n), the Fibonacci numbers, n >= 0 and 0 <= k <= n.
T(n, k) = sum(A035317(n+k-p-2, p), p=0..k), n >= 0 and 0 <= k <= n.
T(n+p+2, p-2) = A080239(n+2*p-1) - sum(A035317(n-k+p-1, k+p-1), k=0..floor(n/2)), n >= 0 and p >= 2.
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) = A226205(k+1).
The two G.f.’s given below generate the terms in the n-th row of the square array Tsq(n, k). The remarkable second G.f. is the partial fraction expansion of the first G.f..
G.f.: 1/((1-x)^(n-2)*(1+x)*(x^2-3*x+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, 0
2| 1, 1, 3
3| 1, 2, 4, 5
4| 1, 3, 6, 9, 16
5| 1, 4, 9, 15, 25, 39
6| 1, 5, 13, 24, 40, 64, 105
7| 1, 6, 18, 37, 64, 104, 169, 272
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, 0, 3, 5, 16, 39, 105, 272
1| 1, 1, 4, 9, 25, 64, 169, 441
2| 1, 2, 6, 15, 40, 104, 273, 714
3| 1, 3, 9, 24, 64, 168, 441, 1155
4| 1, 4, 13, 37, 101, 269, 710, 1865
5| 1, 5, 18, 55, 156, 425, 1135, 3000
6| 1, 6, 24, 79, 235, 660, 1795, 4795
7| 1, 7, 31, 110, 345, 1005, 2800, 7595
MAPLE
T := proc(n, k) option remember: if k=0 then return(1) elif k=n then return(combinat[fibonacci](n+2)*combinat[fibonacci](n-1)) 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.
CROSSREFS
KEYWORD
AUTHOR
Johannes W. Meijer, Oct 19 2013
STATUS
approved