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A026374
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Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.
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18
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1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 11, 6, 1, 1, 7, 17, 17, 7, 1, 1, 9, 30, 45, 30, 9, 1, 1, 10, 39, 75, 75, 39, 10, 1, 1, 12, 58, 144, 195, 144, 58, 12, 1, 1, 13, 70, 202, 339, 339, 202, 70, 13, 1, 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1
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OFFSET
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0,5
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COMMENTS
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T(n,k) is number of lattice paths from (0,0) to (n,n-2k) using steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: T(4,1)=6 because we have the following paths from (0,0) to (4,2): UUUD, UUH, UUDU, UDUU, HUU and DUUU. Row sums yield A026383. Column 1 is A032766, column 2 is A026381, column 3 is A026382. - Emeric Deutsch, Jan 25 2004
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LINKS
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FORMULA
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T(n, k) = number of integer strings s(0), ..., s(n) such that s(0)=0, s(n) = n-2k, where, for 1 <= i <= n, s(i) is even if i is even and |s(i) - s(i-1)| <= 1.
T(2n, k) = Sum_{j=ceiling(k/2)..k} 3^(2j-k)*binomial(n, j)*binomial(j, k-j);
T(2n+1, k) = T(2n, k-1) + T(2n, k).
G.f.: (1 + z + t*z)/(1 - (1+3*t+t^2)*z^2) = 1 + (1+t)*z + (1+3*t+t^2)*z^2+ ... .
Generating polynomial for row 2n is (1 + 3*t + t^2)^n;
Generating polynomial for row 2n+1 it is (1+t)*(1 + 3*t + t^2)^n. (End)
T(2n, k) = Sum_{j=ceiling(k/2)..k} 3^(2j-k)*binomial(n, j)*binomial(j, k-j);
T(2n+1, k) = T(2n, k-1) + T(2n, k). (End)
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EXAMPLE
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Triangle starts:
1;
1, 1;
1, 3, 1;
1, 4, 4, 1;
1, 6, 11, 6, 1;
1, 7, 17, 17, 7, 1;
1, 9, 30, 45, 30, 9, 1;
1, 10, 39, 75, 75, 39, 10, 1;
1, 12, 58, 144, 195, 144, 58, 12, 1;
1, 13, 70, 202, 339, 339, 202, 70, 13, 1;
1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1;
1, 16, 110, 425, 1015, 1558, 1558, 1015, 425, 110, 16, 1;
(End)
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MATHEMATICA
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p[x, 1] := 1;
p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + 1)^Floor[n/2]];
a = Table[CoefficientList[p[x, n], x], {n, 1, 12}];
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PROG
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(Haskell)
a026374 n k = a026374_tabl !! n !! k
a026374_row n = a026374_tabl !! n
a026374_tabl = [1] : map fst (map snd $ iterate f (1, ([1, 1], [1]))) where
f (0, (us, vs)) = (1, (zipWith (+) ([0] ++ us) (us ++ [0]), us))
f (1, (us, vs)) = (0, (zipWith (+) ([0] ++ vs ++ [0]) $
zipWith (+) ([0] ++ us) (us ++ [0]), us))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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