A230449 T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = A052952(n), n >= 0 and 0 <= k <= n.

1, 1, 1, 1, 2, 3, 1, 3, 5, 4, 1, 4, 8, 9, 8, 1, 5, 12, 17, 17, 12, 1, 6, 17, 29, 34, 29, 21, 1, 7, 23, 46, 63, 63, 50, 33, 1, 8, 30, 69, 109, 126, 113, 83, 55, 1, 9, 38, 99, 178, 235, 239, 196, 138, 88, 1, 10, 47, 137, 277, 413, 474, 435, 334, 226, 144

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).

Links

Table of n, a(n) for n=0..65.

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.

Tsq(n, k) = A052952(2*n+k) - sum(A035317(n+k+i+1, k+2*i+2), i = 0..n-1)

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.
T := proc(n, k): add(A035317(k-p+n-k, k-2*p), p=0..floor(k/2)) end: A035317 := proc(n, k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(T(n, k), k=0..n), n=0..10); # End second program.

Crossrefs

Cf. (Triangle columns) A000012, A000027, A089071, A052952, A129696

Cf. A230447, A230448

Sequence in context: A121775 A127951 A208814 * A255437 A162609 A194752

Adjacent sequences: A230446 A230447 A230448 * A230450 A230451 A230452

Keyword

nonn,easy,tabl

Author

Johannes W. Meijer, Oct 19 2013

Status

approved