This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027926 #35 Sep 08 2022 08:44:49
%S A027926 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,
%T A027926 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,
%U A027926 22,7,1,1,1,2,3,5,8,13,21,34,54,79,97,92,63,29,8,1
%N 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.
%C A027926 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.
%C A027926 Can be seen as concatenation of triangles A104763 and A105809, with identifying column of Fibonacci numbers, see example. - _Reinhard Zumkeller_, Aug 15 2013
%H A027926 Reinhard Zumkeller, <a href="/A027926/b027926.txt">Rows n = 0..100 of table, flattened</a>
%H A027926 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A027926 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
%e A027926 . 0: 1
%e A027926 . 1: 1 1 1
%e A027926 . 2: 1 1 2 2 1
%e A027926 . 3: 1 1 2 3 4 3 1
%e A027926 . 4: 1 1 2 3 5 7 7 4 1
%e A027926 . 5: 1 1 2 3 5 8 12 14 11 5 1
%e A027926 . 6: 1 1 2 3 5 8 13 20 26 25 16 6 1
%e A027926 . 7: 1 1 2 3 5 8 13 21 33 46 51 41 22 7 1
%e A027926 . 8: 1 1 2 3 5 8 13 21 34 54 79 97 92 63 29 8 1
%e A027926 . 9: 1 1 2 3 5 8 13 21 34 55 88 133 176 189 155 92 37 9 1
%e A027926 . 10: 1 1 2 3 5 8 13 21 34 55 89 143 221 309 365 344 247 129 46 10 1
%e A027926 .
%e A027926 . 1: 1
%e A027926 . 2: 1 1
%e A027926 . 3: 1 1 2
%e A027926 . 4: 1 1 2 3
%e A027926 . 5: 1 1 2 3 5 columns = A000045, > 0
%e A027926 . 6: 1 1 2 3 5 8 +---------+
%e A027926 . 7: 1 1 2 3 5 8 13 | A104763 |
%e A027926 . 8: 1 1 2 3 5 8 13 21 +---------+
%e A027926 . 9: 1 1 2 3 5 8 13 21 34
%e A027926 . 10: 1 1 2 3 5 8 13 21 34 55
%e A027926 . 11: 1 1 2 3 5 8 13 21 34 55 89
%e A027926 .
%e A027926 . 0: 1
%e A027926 . 1: 1 1 +---------+
%e A027926 . 2: 2 2 1 | A105809 |
%e A027926 . 3: 3 4 3 1 +---------+
%e A027926 . 4: 5 7 7 4 1
%e A027926 . 5: 8 12 14 11 5 1
%e A027926 . 6: 13 20 26 25 16 6 1
%e A027926 . 7: 21 33 46 51 41 22 7 1
%e A027926 . 8: 34 54 79 97 92 63 29 8 1
%e A027926 . 9: 55 88 133 176 189 155 92 37 9 1
%e A027926 . 10: 89 143 221 309 365 344 247 129 46 10 1
%p A027926 A027926 := proc(n,k)
%p A027926 add(binomial(n-j,2*n-k-2*j),j=0..(2*n-k+1)/2) ;
%p A027926 end proc: # _R. J. Mathar_, Apr 11 2016
%t A027926 z = 15; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := 1;
%t A027926 t[n_, k_] := t[n, k] = t[n - 1, k - 2] + t[n - 1, k - 1];
%t A027926 u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
%t A027926 TableForm[u] (* A027926 array *)
%t A027926 v = Flatten[u] (* A027926 sequence *)
%t A027926 (* _Clark Kimberling_, Aug 31 2014 *)
%t A027926 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 *)
%o A027926 (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 */
%o A027926 (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_ */
%o A027926 (Haskell)
%o A027926 a027926 n k = a027926_tabf !! n !! k
%o A027926 a027926_row n = a027926_tabf !! n
%o A027926 a027926_tabf = iterate (\xs -> zipWith (+)
%o A027926 ([0] ++ xs ++ [0]) ([1,0] ++ xs)) [1]
%o A027926 -- Variant, cf. example:
%o A027926 a027926_tabf' = zipWith (++) a104763_tabl (map tail a105809_tabl)
%o A027926 -- _Reinhard Zumkeller_, Aug 15 2013
%o A027926 (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
%o A027926 (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
%o A027926 (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
%Y A027926 Many columns of T are A000045 (Fibonacci sequence), also in T: A001924, A004006, A000071, A000124, A014162, A014166, A027927-A027933.
%Y A027926 Some other Fibonacci-Pascal triangles: A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074.
%K A027926 nonn,tabf
%O A027926 0,7
%A A027926 _Clark Kimberling_
%E A027926 Incorporates comments from _Michael Somos_.
%E A027926 Example extended by _Reinhard Zumkeller_, Aug 15 2013