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%I #4 Oct 19 2013 10:46:06
%S 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,
%T 23,46,63,63,50,33,1,8,30,69,109,126,113,83,55,1,9,38,99,178,235,239,
%U 196,138,88,1,10,47,137,277,413,474,435,334,226,144
%N 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.
%C 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.
%C The row sums lead to A094687, the convolution of Fibonacci and Jacobsthal numbers, and the alternating row sums lead to A008346.
%C The backwards antidiagonal sums equal Kn21(n) = (-1)^n*A175722(n).
%F 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.
%F T(n+p-1, n) = sum(A035317(n-k+p-1, n-2*k), k=0..floor(n/2)), n >= 0 and p >= 1.
%F The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
%F Tsq(n, k) = sum(Tsq(n-1, i), i=0..k), n >= 1 and k >= 0, with Tsq(0, k) = A052952(k).
%F Tsq(n, k) = sum(A035317(n+k-i, k-2*i), i=0..floor(k/2)), n >= 0 and k >= 0.
%F Tsq(n, k) = A052952(2*n+k) - sum(A035317(n+k+i+1, k+2*i+2), i = 0..n-1)
%F The G.f. generates the terms in the n-th row of the square array Tsq(n, k).
%F G.f.: (-1)^(n)/((-1+x+x^2)*(x+1)*(x-1)^(n+1)), n >= 0.
%e The first few rows of triangle T(n, k), n >= 0 and 0 <= k <= n.
%e n/k 0 1 2 3 4 5 6 7
%e ------------------------------------------------
%e 0| 1
%e 1| 1, 1
%e 2| 1, 2, 3
%e 3| 1, 3, 5, 4
%e 4| 1, 4, 8, 9, 8
%e 5| 1, 5, 12, 17, 17, 12
%e 6| 1, 6, 17, 29, 34, 29, 21
%e 7| 1, 7, 23, 46, 63, 63, 50, 33
%e The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
%e n/k 0 1 2 3 4 5 6 7
%e ------------------------------------------------
%e 0| 1, 1, 3, 4, 8, 12, 21, 33
%e 1| 1, 2, 5, 9, 17, 29, 50, 83
%e 2| 1, 3, 8, 17, 34, 63, 113, 196
%e 3| 1, 4, 12, 29, 63, 126, 239, 435
%e 4| 1, 5, 17, 46, 109, 235, 474, 909
%e 5| 1, 6, 23, 69, 178, 413, 887, 1796
%e 6| 1, 7, 30, 99, 277, 690, 1577, 3373
%e 7| 1, 8, 38, 137, 414, 1104, 2681, 6054
%p 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.
%p 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.
%Y Cf. (Triangle columns) A000012, A000027, A089071, A052952, A129696
%Y Cf. A230447, A230448
%K nonn,easy,tabl
%O 0,5
%A _Johannes W. Meijer_, Oct 19 2013
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