A230447 T(n, k) = T(n-1, k) + T(n-1, k-1) + A230135(n, k) with T(n, 0) = A008619(n) and T(n, n) = A080239(n+1), n >= 0 and 0 <= k <= n.

1, 1, 1, 2, 2, 2, 2, 4, 5, 3, 3, 6, 9, 8, 6, 3, 9, 16, 17, 14, 9, 4, 12, 25, 33, 32, 23, 15, 4, 16, 38, 58, 65, 55, 39, 24, 5, 20, 54, 96, 124, 120, 94, 63, 40, 5, 25, 75, 150, 220, 244, 215, 157, 103, 64, 6, 30, 100, 225, 371, 464, 459, 372, 261, 167, 104

Offset: 0

Author

Johannes W. Meijer, Oct 19 2013

Keywords

• nonn
• easy
• tabl

Comments

The terms in the right hand columns of triangle T(n, k) and the terms in the rows of the square array Tsq(n, k) represent the Kn1p sums of the ‘Races with Ties’ triangle A035317.

For the definitions of the Kn1p sums see A180662. This sequence is related to A230448.

The first few row sums are: 1, 2, 6, 14, 32, 68, 144, 299, 616, 1258, 2559, 5185, 10478, … .

Examples

			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|  2,  2,  2
3|  2,  4,  5,   3
4|  3,  6,  9,   8,   6
5|  3,  9, 16,  17,  14,    9
6|  4, 12, 25,  33,  32,   23,    15
7|  4, 16, 38,  58,  65,   55,    39,   24
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,  2,   3,   6,    9,   15,   24
1|  1,  2,  5,   8,  14,   23,   39,   63
2|  2,  4,  9,  17,  32,   55,   94,  157
3|  2,  6, 16,  33,  65,  120,  215,  372
4|  3,  9, 25,  58, 124,  244,  459,  831
5|  3, 12, 38,  96, 220,  464,  924, 1755
6|  4, 16, 54, 150, 371,  835, 1759, 3514
7|  4, 20, 75, 225, 596, 1431, 3191, 6705
		

Crossrefs

Cf. (Triangle columns) A008619, A002620, A175287, A080239

Cf. A035317, A230448, A230449, A230135, A080239, A034851, A228570

Programs

Maple

T := proc(n, k): add(A035317(n-i, n-k+i), i=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 first program.
T := proc(n, k) option remember: if k=0 then return(A008619(n)) elif k=n then return(A080239(n+1)) else A230135(n, k) + procname(n-1, k) + procname(n-1, k-1) fi: end: A008619 := n -> floor(n/2) +1: A080239 := n -> add(combinat[fibonacci](n-4*k), k=0..floor((n-1)/4)): A230135 := proc(n, k): if ((k mod 4 = 2) and (n mod 2 = 1)) or ((k mod 4 = 0) and (n mod 2 = 0)) then return(1) else return(0) fi: end: seq(seq(T(n, k), k=0..n), n=0..10); # End second program.

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + A230135(n, k) with T(n, 0) = A008619(n) and T(n, n) = A080239(n+1), n >= 0 and 0 <= k <= n.

T(n, k) = sum(A035317(n-i, n-k+i), i = 0..floor(k/2)), n >= 0 and 0 <= k <= n.

The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.

Tsq(n, k) = sum(A035317(n+k-i, n+i), i=0..floor(k/2)), n >= 0 and k >= 0.

Tsq(n, k) = A080239(2*n+k+1) - sum(A035317(2*n+k-i, i), i=0..n-1).

The G.f. generates the terms in the n-th row of the square array Tsq(n, k).

G.f.: a(n)/(4*(x-1)) + 1/(4*(x+1)) + (-1)^n*(x+2)/(10*(x^2+1)) - (A000032(2*n+3) + A000032(2*n+2)*x)/(5*(x^2+x-1)) + sum((-1)^(k+1) * A064831(n-k+1)/((x-1)^k), k= 2..n), n >= 0, with a(n) = A064831(n+1) + 2*A064831(n) - 2*A064831(n-1) + A064831(n-2).