cp's OEIS Frontend

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.

%I A230447 #6 Oct 19 2013 10:45:34
%S A230447 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 A230447 16,38,58,65,55,39,24,5,20,54,96,124,120,94,63,40,5,25,75,150,220,244,
%U A230447 215,157,103,64,6,30,100,225,371,464,459,372,261,167,104
%N 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.
%C A230447 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 A230447 For the definitions of the Kn1p sums see A180662. This sequence is related to A230448.
%C A230447 The first few row sums are: 1, 2, 6, 14, 32, 68, 144, 299, 616, 1258, 2559, 5185, 10478, … .
%F 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.
%F A230447 T(n, k) = sum(A035317(n-i, n-k+i), i = 0..floor(k/2)), n >= 0 and 0 <= k <= n.
%F A230447 The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
%F A230447 Tsq(n, k) = sum(A035317(n+k-i, n+i), i=0..floor(k/2)), n >= 0 and k >= 0.
%F A230447 Tsq(n, k) = A080239(2*n+k+1) - sum(A035317(2*n+k-i, i), i=0..n-1).
%F A230447 The G.f. generates the terms in the n-th row of the square array Tsq(n, k).
%F A230447 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 A230447 The first few rows of triangle T(n, k) n >= 0 and 0 <= k <= n.
%e A230447 n/k 0   1   2    3    4     5     6     7
%e A230447 ------------------------------------------------
%e A230447 0|  1
%e A230447 1|  1,  1
%e A230447 2|  2,  2,  2
%e A230447 3|  2,  4,  5,   3
%e A230447 4|  3,  6,  9,   8,   6
%e A230447 5|  3,  9, 16,  17,  14,    9
%e A230447 6|  4, 12, 25,  33,  32,   23,    15
%e A230447 7|  4, 16, 38,  58,  65,   55,    39,   24
%e A230447 The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
%e A230447 n/k 0   1   2    3    4     5     6     7
%e A230447 ------------------------------------------------
%e A230447 0|  1,  1,  2,   3,   6,    9,   15,   24
%e A230447 1|  1,  2,  5,   8,  14,   23,   39,   63
%e A230447 2|  2,  4,  9,  17,  32,   55,   94,  157
%e A230447 3|  2,  6, 16,  33,  65,  120,  215,  372
%e A230447 4|  3,  9, 25,  58, 124,  244,  459,  831
%e A230447 5|  3, 12, 38,  96, 220,  464,  924, 1755
%e A230447 6|  4, 16, 54, 150, 371,  835, 1759, 3514
%e A230447 7|  4, 20, 75, 225, 596, 1431, 3191, 6705
%p A230447 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 A230447 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 A230447 Cf. (Triangle columns) A008619, A002620, A175287, A080239
%Y A230447 Cf. A035317, A230448, A230449, A230135, A080239, A034851, A228570
%K A230447 nonn,easy,tabl
%O A230447 0,4
%A A230447 _Johannes W. Meijer_, Oct 19 2013