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 A035317 Pascal-like triangle associated with A000670. 24
 1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 4, 7, 6, 3, 1, 5, 11, 13, 9, 3, 1, 6, 16, 24, 22, 12, 4, 1, 7, 22, 40, 46, 34, 16, 4, 1, 8, 29, 62, 86, 80, 50, 20, 5, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 1, 11, 56, 174, 367, 553, 610, 496, 295, 125 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS From Johannes W. Meijer, Jul 20 2011: (Start) The triangle sums, see A180662 for their definitions, link this "Races with Ties" triangle with several sequences, see the crossrefs. Observe that the Kn4 sums lead to the golden rectangle numbers A001654 and that the Fi1 and Fi2 sums lead to the Jacobsthal sequence A001045. The series expansion of G(x, y) = 1/((y*x-1)*(y*x+1)*((y+1)*x-1)) as function of x leads to this sequence, see the second Maple program. (End) T(2*n+1,n) = A014301(n+1); T(2*n+1,n+1) = A026641(n+1). - Reinhard Zumkeller, Jul 19 2012 LINKS Vincenzo Librandi, Rows n = 0..100, flattened A. Hlavác, M. Marvan, Nonlocal conservation laws of the constant astigmatism equation, arXiv preprint arXiv:1602.06861 [nlin.SI], 2016. E. Mendelson, Races with Ties, Math. Mag. 55 (1982), 170-175. FORMULA T(n,k) = Sum_{j=0..floor(n/2)} binomial(n-2j, k-2j). - Paul Barry, Feb 11 2003 From Johannes W. Meijer, Jul 20 2011: (Start) T(n, k) = Sum_{i=0..k}((-1)^(i+k) * binomial(i+n-k+1,i)). (Mendelson) T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = floor(n/2) + 1. (Mendelson) Sum_{k = 0..n}((-1)^k * (n-k+1)^n * T(n, k)) = A000670(n). (Mendelson) T(n, n-k) = A128176(n, k); T(n+k, n-k) = A158909(n, k); T(2*n-k, k) = A092879(n, k). (End) EXAMPLE Triangle begins:   1;   1,  1;   1,  2,  2;   1,  3,  4,   2;   1,  4,  7,   6,   3;   1,  5, 11,  13,   9,   3;   1,  6, 16,  24,  22,  12,   4;   1,  7, 22,  40,  46,  34,  16,   4;   1,  8, 29,  62,  86,  80,  50,  20,  5;   1,  9, 37,  91, 148, 166, 130,  70, 25,  5;   1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6;   ... MAPLE A035317 := proc(n, k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(A035317(n, k), k=0..n), n=0..10); # Johannes W. Meijer, Jul 20 2011 A035317 := proc(n, k): coeff(coeftayl(1/((y*x-1)*(y*x+1)*((y+1)*x-1)), x=0, n), y, k) end: seq(seq(A035317(n, k), k=0..n), n=0..10); # Johannes W. Meijer, Jul 20 2011 MATHEMATICA t[n_, k_] := (-1)^k*(((-1)^k*(n+2)!*Hypergeometric2F1[1, n+3, k+2, -1])/((k+1)!*(n-k+1)!) + 2^(k-n-2)); Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Dec 14 2011, after Johannes W. Meijer *) PROG (Haskell) a035317 n k = a035317_tabl !! n !! k a035317_row n = a035317_tabl !! n a035317_tabl = map snd \$ iterate f (0, [1]) where    f (i, row) = (1 - i, zipWith (+) ([0] ++ row) (row ++ [i])) -- Reinhard Zumkeller, Jul 09 2012 (PARI) {T(n, k)=if(n==k, (n+2)\2, if(k==0, 1, if(n>k, T(n-1, k-1)+T(n-1, k))))} for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jul 18 2012 (Sage) def A035317_row(n):     @cached_function     def prec(n, k):         if k==n: return 1         if k==0: return 0         return -prec(n-1, k-1)-sum(prec(n, k+i-1) for i in (2..n-k+1))     return [(-1)^k*prec(n+2, k) for k in (1..n)] for n in (1..11): print(A035317_row(n)) # Peter Luschny, Mar 16 2016 CROSSREFS Row sums are A000975, diagonal sums are A080239. Central terms are A014300. Similar to the triangles A059259, A080242, A108561, A112555. Cf. A059260. Triangle sums (see the comments): A000975 (Row1), A059841 (Row2), A080239 (Kn11), A052952 (Kn21), A129696 (Kn22), A001906 (Kn3), A001654 (Kn4), A001045 (Fi1, Fi2), A023435 (Ca2), Gi2 (A193146), A190525 (Ze2), A193147 (Ze3), A181532 (Ze4). - Johannes W. Meijer, Jul 20 2011 Cf. A181971. Sequence in context: A339708 A080242 A183927 * A103923 A186711 A061987 Adjacent sequences:  A035314 A035315 A035316 * A035318 A035319 A035320 KEYWORD nonn,easy,tabl,nice AUTHOR EXTENSIONS More terms from James A. Sellers STATUS approved

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Last modified May 7 20:36 EDT 2021. Contains 343652 sequences. (Running on oeis4.)