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%I #6 Oct 19 2013 10:45:34
%S 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,
%T 16,38,58,65,55,39,24,5,20,54,96,124,120,94,63,40,5,25,75,150,220,244,
%U 215,157,103,64,6,30,100,225,371,464,459,372,261,167,104
%N 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.
%C 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.
%C For the definitions of the Kn1p sums see A180662. This sequence is related to A230448.
%C The first few row sums are: 1, 2, 6, 14, 32, 68, 144, 299, 616, 1258, 2559, 5185, 10478, … .
%F 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.
%F T(n, k) = sum(A035317(n-i, n-k+i), i = 0..floor(k/2)), n >= 0 and 0 <= k <= n.
%F The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
%F Tsq(n, k) = sum(A035317(n+k-i, n+i), i=0..floor(k/2)), n >= 0 and k >= 0.
%F Tsq(n, k) = A080239(2*n+k+1) - sum(A035317(2*n+k-i, i), 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.: 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).
%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| 2, 2, 2
%e 3| 2, 4, 5, 3
%e 4| 3, 6, 9, 8, 6
%e 5| 3, 9, 16, 17, 14, 9
%e 6| 4, 12, 25, 33, 32, 23, 15
%e 7| 4, 16, 38, 58, 65, 55, 39, 24
%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, 2, 3, 6, 9, 15, 24
%e 1| 1, 2, 5, 8, 14, 23, 39, 63
%e 2| 2, 4, 9, 17, 32, 55, 94, 157
%e 3| 2, 6, 16, 33, 65, 120, 215, 372
%e 4| 3, 9, 25, 58, 124, 244, 459, 831
%e 5| 3, 12, 38, 96, 220, 464, 924, 1755
%e 6| 4, 16, 54, 150, 371, 835, 1759, 3514
%e 7| 4, 20, 75, 225, 596, 1431, 3191, 6705
%p 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.
%p 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.
%Y Cf. (Triangle columns) A008619, A002620, A175287, A080239
%Y Cf. A035317, A230448, A230449, A230135, A080239, A034851, A228570
%K nonn,easy,tabl
%O 0,4
%A _Johannes W. Meijer_, Oct 19 2013
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