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A181971 Triangle read by rows: T(n,0) = 1, T(n,n) = floor((n+3)/2) and T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.

%I A181971 #16 Oct 12 2021 08:01:40
%S A181971 1,1,2,1,3,2,1,4,5,3,1,5,9,8,3,1,6,14,17,11,4,1,7,20,31,28,15,4,1,8,
%T A181971 27,51,59,43,19,5,1,9,35,78,110,102,62,24,5,1,10,44,113,188,212,164,
%U A181971 86,29,6,1,11,54,157,301,400,376,250,115,35,6,1,12,65,211,458,701,776,626,365,150,41,7
%N A181971 Triangle read by rows: T(n,0) = 1, T(n,n) = floor((n+3)/2) and T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.
%C A181971 Another variant of Pascal's triangle;
%C A181971 row sums: A081254; central terms: T(2*n,n) = A128082(n+1);
%C A181971 T(n,0) = 1;
%C A181971 T(n,1) = n + 1 for n > 0;
%C A181971 T(n,2) = A000096(n-1) for n > 1;
%C A181971 T(n,3) = A105163(n-2) for n > 2;
%C A181971 T(n,n-2) = A005744(n-1) for n > 1;
%C A181971 T(n,n-1) = A024206(n) for n > 0;
%C A181971 T(n,n) = A008619(n+1).
%H A181971 Reinhard Zumkeller, <a href="/A181971/b181971.txt">Rows n=0..150 of triangle, flattened</a>
%H A181971 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%e A181971 The triangle begins:
%e A181971 .  0:                              1
%e A181971 .  1:                           1     2
%e A181971 .  2:                        1     3     2
%e A181971 .  3:                     1     4     5     3
%e A181971 .  4:                  1     5     9     8     3
%e A181971 .  5:               1     6    14    17    11     4
%e A181971 .  6:            1     7    20    31    28    15     4
%e A181971 .  7:         1     8    27    51    59    43    19     5
%e A181971 .  8:      1     9    35    78   110   102    62    24     5
%e A181971 .  9:   1    10    44   113   188   212   164    86    29     6.
%t A181971 T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = If[n == k, Quotient[n + 3, 2], If[k == 0, 1, If[n > k, T[n - 1, k - 1] + T[n - 1, k]]]];
%t A181971 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 12 2021 *)
%o A181971 (Haskell)
%o A181971 a181971 n k = a181971_tabl !! n !! k
%o A181971 a181971_row n = a181971_tabl !! n
%o A181971 a181971_tabl = map snd $ iterate f (1, [1]) where
%o A181971    f (i, row) = (1 - i, zipWith (+) ([0] ++ row) (row ++ [i]))
%o A181971 (PARI) {T(n,k)=if(n==k,(n+3)\2,if(k==0,1,if(n>k,T(n-1,k-1)+T(n-1,k))))}
%o A181971 for(n=0,12,for(k=0,n,print1(T(n,k),","));print("")) \\ _Paul D. Hanna_, Jul 18 2012
%Y A181971 Cf. A035317, A007318, A074909.
%K A181971 nonn,tabl
%O A181971 0,3
%A A181971 _Reinhard Zumkeller_, Jul 09 2012