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 A027926 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. 44
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS 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. Can be seen as concatenation of triangles A104763 and A105809, with identifying column of Fibonacci numbers, see example. - Reinhard Zumkeller, Aug 15 2013 LINKS Reinhard Zumkeller, Rows n = 0..100 of table, flattened FORMULA 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 EXAMPLE .   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     +---------+ .   7:          1 1  2  3  5  8 13     | A104763 | .   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 . .   0:                           1 .   1:                           1   1                +---------+ .   2:                           2   2   1            | A105809 | .   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 MAPLE A027926 := proc(n, k)     add(binomial(n-j, 2*n-k-2*j), j=0..(2*n-k+1)/2) ; end proc: # R. J. Mathar, Apr 11 2016 MATHEMATICA 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}]; TableForm[u] (* A027926 array *) v = Flatten[u] (* A027926 sequence *) (* Clark Kimberling, Aug 31 2014 *) 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 *) PROG (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) -- Reinhard Zumkeller, Aug 15 2013 (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 CROSSREFS Many columns of T are A000045 (Fibonacci sequence), also in T: A001924, A004006, A000071, A000124, A014162, A014166, A027927-A027933. Some other Fibonacci-Pascal triangles: A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074. Sequence in context: A071127 A029381 A297877 * A114730 A031282 A085685 Adjacent sequences:  A027923 A027924 A027925 * A027927 A027928 A027929 KEYWORD nonn,tabf AUTHOR EXTENSIONS Incorporates comments from Michael Somos. Example extended by Reinhard Zumkeller, Aug 15 2013 STATUS approved

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Last modified October 21 04:40 EDT 2019. Contains 328291 sequences. (Running on oeis4.)