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 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified July 23 22:20 EDT 2021. Contains 346265 sequences. (Running on oeis4.)