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A035317
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Pascal-like triangle associated with A000670.
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24
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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
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table;
graph;
refs;
listen;
history;
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OFFSET
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0,5
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COMMENTS
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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)
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LINKS
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FORMULA
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T(n,k) = Sum_{j=0..floor(n/2)} binomial(n-2j, k-2j). - Paul Barry, Feb 11 2003
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)
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EXAMPLE
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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;
...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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]))
(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)
@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)]
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CROSSREFS
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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
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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