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A134636 Triangle formed by Pascal's rule given borders = 2n+1.

%I A134636 #28 Mar 26 2022 03:58:20
%S A134636 1,3,3,5,6,5,7,11,11,7,9,18,22,18,9,11,27,40,40,27,11,13,38,67,80,67,
%T A134636 38,13,15,51,105,147,147,105,51,15,17,66,156,252,294,252,156,66,17,19,
%U A134636 83,222,408,546,546,408,222,83,19,21,102,305,630,954,1092,954,630,305,102,21
%N A134636 Triangle formed by Pascal's rule given borders = 2n+1.
%C A134636 Row sums = A048487: (1, 6, 16, 36, 76, 156, ...).
%H A134636 Reinhard Zumkeller, <a href="/A134636/b134636.txt">Rows n = 0..120 of triangle, flattened</a>
%H A134636 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A134636 Triangle, given borders = (1, 3, 5, 7, 9, ...); apply Pascal's rule T(n,k) = T(n-1,k) P T(n-1,k-1).
%F A134636 T(n,k) = A051601(n,k) + A051597(n,k); T(n,k) mod 2 = A047999(n,k). - _Reinhard Zumkeller_, Nov 23 2012
%F A134636 Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - _Boris Putievskiy_, Aug 19 2013
%e A134636 First few rows of the triangle:
%e A134636    1;
%e A134636    3,  3;
%e A134636    5,  6,  5;
%e A134636    7, 11, 11,  7;
%e A134636    9, 18, 22, 18,  9;
%e A134636   11, 27, 40, 40, 27, 11;
%e A134636   13, 38, 67, 80, 67, 38, 13;
%e A134636   ...
%p A134636 T:= proc(n,k) option remember;
%p A134636       `if`(k<0 or k>n, 0,
%p A134636       `if`(k=0 or k=n, 2*n+1,
%p A134636          T(n-1, k-1) + T(n-1, k) ))
%p A134636     end:
%p A134636 seq(seq(T(n, k), k=0..n), n=0..14);  # _Alois P. Heinz_, May 26 2013
%t A134636 NestList[Append[Prepend[Map[Apply[Plus, #] &, Partition[#, 2, 1]], #[[1]] + 2], #[[1]] + 2] &, {1}, 10] // Grid  (* _Geoffrey Critzer_, May 26 2013 *)
%t A134636 T[n_, k_] := Binomial[n, k-1] + Binomial[n, k] + 2 Binomial[n, k+1] + Binomial[n, n-k+1];
%t A134636 Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 07 2021 *)
%o A134636 (Haskell)
%o A134636 a134636 n k = a134636_tabl !! n !! k
%o A134636 a134636_row n = a134636_tabl !! n
%o A134636 a134636_tabl = iterate (\row -> zipWith (+) ([2] ++ row) (row ++ [2])) [1]
%o A134636 -- _Reinhard Zumkeller_, Nov 23 2012
%Y A134636 Cf. A007318, A048487, A051601, A051597.
%K A134636 nonn,tabl
%O A134636 0,2
%A A134636 _Gary W. Adamson_, Nov 04 2007
%E A134636 Offset changed by _Reinhard Zumkeller_, Nov 23 2012