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A027926
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Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2.
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44
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1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 1, 1, 2, 3, 5, 8, 13, 20, 26, 25, 16, 6, 1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 46, 51, 41, 22, 7, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 79, 97, 92, 63, 29, 8, 1
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OFFSET
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0,7
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COMMENTS
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T(n,k) = number of strings s(0),...,s(n) such that s(0)=0, s(n)=n-k and for 1<=i<=n, s(i)=s(i-1)+d, with d in {0,1,2} if i=0, in {0,2} if s(i)=2i, in {0,1,2} if s(i)=2i-1, in {0,1} if 0<=s(i)<=2i-2.
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..floor((2*n-k+1)/2)} binomial(n-j, 2*n-k-2*j). - Len Smiley, Oct 21 2001
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EXAMPLE
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. 0: 1
. 1: 1 1 1
. 2: 1 1 2 2 1
. 3: 1 1 2 3 4 3 1
. 4: 1 1 2 3 5 7 7 4 1
. 5: 1 1 2 3 5 8 12 14 11 5 1
. 6: 1 1 2 3 5 8 13 20 26 25 16 6 1
. 7: 1 1 2 3 5 8 13 21 33 46 51 41 22 7 1
. 8: 1 1 2 3 5 8 13 21 34 54 79 97 92 63 29 8 1
. 9: 1 1 2 3 5 8 13 21 34 55 88 133 176 189 155 92 37 9 1
. 10: 1 1 2 3 5 8 13 21 34 55 89 143 221 309 365 344 247 129 46 10 1
.
. 1: 1
. 2: 1 1
. 3: 1 1 2
. 4: 1 1 2 3
. 5: 1 1 2 3 5 columns = A000045, > 0
. 6: 1 1 2 3 5 8 +---------+
. 8: 1 1 2 3 5 8 13 21 +---------+
. 9: 1 1 2 3 5 8 13 21 34
. 10: 1 1 2 3 5 8 13 21 34 55
. 11: 1 1 2 3 5 8 13 21 34 55 89
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. 0: 1
. 1: 1 1 +---------+
. 3: 3 4 3 1 +---------+
. 4: 5 7 7 4 1
. 5: 8 12 14 11 5 1
. 6: 13 20 26 25 16 6 1
. 7: 21 33 46 51 41 22 7 1
. 8: 34 54 79 97 92 63 29 8 1
. 9: 55 88 133 176 189 155 92 37 9 1
. 10: 89 143 221 309 365 344 247 129 46 10 1
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MAPLE
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add(binomial(n-j, 2*n-k-2*j), j=0..(2*n-k+1)/2) ;
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MATHEMATICA
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z = 15; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := 1;
t[n_, k_] := t[n, k] = t[n - 1, k - 2] + t[n - 1, k - 1];
u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
v = Flatten[u] (* A027926 sequence *)
Table[Sum[Binomial[n-j, 2*n-k-2*j], {j, 0, Floor[(2*n-k+1)/2]}], {n, 0, 10}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Sep 05 2019 *)
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PROG
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(PARI) {T(n, k) = if( k<0 || k>2*n, 0, if( k<=1 || k==2*n, 1, T(n-1, k-2) + T(n-1, k-1)))}; /* _Michael Somos, Feb 26 1999 */
(PARI) {T(n, k) = if( k<0 || k>2*n, 0, sum( j=max(0, k-n), k\2, binomial(k-j, j)))}; /* Michael Somos */
(Haskell)
a027926 n k = a027926_tabf !! n !! k
a027926_row n = a027926_tabf !! n
a027926_tabf = iterate (\xs -> zipWith (+)
([0] ++ xs ++ [0]) ([1, 0] ++ xs)) [1]
-- Variant, cf. example:
a027926_tabf' = zipWith (++) a104763_tabl (map tail a105809_tabl)
(Magma) [&+[Binomial(n-j, 2*n-k-2*j): j in [0..Floor((2*n-k+1)/2)]]: k in [0..2*n], n in [0..10]]; // G. C. Greubel, Sep 05 2019
(Sage) [[sum(binomial(n-j, 2*n-k-2*j) for j in (0..floor((2*n-k+1)/2))) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Sep 05 2019
(GAP) Flat(List([0..10], n-> List([0..2*n], k-> Sum([0..Int((2*n-k+1)/2) ], j-> Binomial(n-j, 2*n-k-2*j) )))); # G. C. Greubel, Sep 05 2019
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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