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 A230448 T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = A226205(n+1), n >= 0 and 0 <= k <= n. 2
 1, 1, 0, 1, 1, 3, 1, 2, 4, 5, 1, 3, 6, 9, 16, 1, 4, 9, 15, 25, 39, 1, 5, 13, 24, 40, 64, 105, 1, 6, 18, 37, 64, 104, 169, 272, 1, 7, 24, 55, 101, 168, 273, 441, 715, 1, 8, 31, 79, 156, 269, 441, 714, 1156, 1869, 1, 9, 39, 110, 235, 425, 710, 1155, 1870, 3025, 4896 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Triangle T(n, k) is related to the Kn1p sums of the ‘Races with Ties’ triangle A035317. See A230447 for the Kn1p sums and A180662 for the definitions of these sums. The row sums equal ((-1)^n*3*A083581(n) + A022379(2*n+2))/15. Note that the partial fraction expansion of the G.f. of the terms in the n-th row of the square array Tsq(n, k) = T(n+k, k) is related to A014334, the exponential convolution of the Fibonacci numbers with themselves, and to A000032, the Lucas numbers. 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) * F(n-1) = A226205(n+1) with F(n) = A000045(n), the Fibonacci numbers, n >= 0 and 0 <= k <= n. T(n, k) = sum(A035317(n+k-p-2, p), p=0..k), n >= 0 and 0 <= k <= n. T(n+p+2, p-2) = A080239(n+2*p-1) - sum(A035317(n-k+p-1, k+p-1), k=0..floor(n/2)), n >= 0 and p >= 2. 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) = A226205(k+1). The two G.f.’s given below generate the terms in the n-th row of the square array Tsq(n, k). The remarkable second G.f. is the partial fraction expansion of the first G.f.. G.f.: 1/((1-x)^(n-2)*(1+x)*(x^2-3*x+1)), n >= 0. G.f.: sum((-1)^(n+k-1)*A014334(k+2)/(2^(k+2)*(x-1)^(n-k-2)), k=0..n-3) + 1/(5*2^(n-2)*(1+x)) + (A000032(n+1) - A000032(n-1)*x)/(5*(x^2-3*x+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,  0 2|  1,  1,  3 3|  1,  2,  4,   5 4|  1,  3,  6,   9,  16 5|  1,  4,  9,  15,  25,   39 6|  1,  5, 13,  24,  40,   64,  105 7|  1,  6, 18,  37,  64,  104,  169,   272 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,  0,  3,   5,  16,  39,   105,  272 1|  1,  1,  4,   9,  25,  64,   169,  441 2|  1,  2,  6,  15,  40,  104,  273,  714 3|  1,  3,  9,  24,  64,  168,  441, 1155 4|  1,  4, 13,  37, 101,  269,  710, 1865 5|  1,  5, 18,  55, 156,  425, 1135, 3000 6|  1,  6, 24,  79, 235,  660, 1795, 4795 7|  1,  7, 31, 110, 345, 1005, 2800, 7595 MAPLE T := proc(n, k) option remember: if k=0 then return(1) elif k=n then return(combinat[fibonacci](n+2)*combinat[fibonacci](n-1)) 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(n+k-p-2, p), p=0..k) 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, A001477, A152950, A226205, A007598, A001654, A064831, A080145 Sequence in context: A183312 A108038 A151845 * A201653 A230765 A250306 Adjacent sequences:  A230445 A230446 A230447 * A230449 A230450 A230451 KEYWORD nonn,easy,tabl AUTHOR Johannes W. Meijer, Oct 19 2013 STATUS approved

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Last modified September 20 17:42 EDT 2021. Contains 347588 sequences. (Running on oeis4.)