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.

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, 27, 51, 59, 43, 19, 5, 1, 9, 35, 78, 110, 102, 62, 24, 5, 1, 10, 44, 113, 188, 212, 164, 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

Offset: 0

Author

Reinhard Zumkeller, Jul 09 2012

Keywords

• nonn
• tabl

Comments

Another variant of Pascal's triangle;

row sums: A081254; central terms: T(2*n,n) = A128082(n+1);

T(n,0) = 1;

T(n,1) = n + 1 for n > 0;

T(n,2) = A000096(n-1) for n > 1;

T(n,3) = A105163(n-2) for n > 2;

T(n,n-2) = A005744(n-1) for n > 1;

T(n,n-1) = A024206(n) for n > 0;

T(n,n) = A008619(n+1).

Examples

			The triangle begins:
.  0:                              1
.  1:                           1     2
.  2:                        1     3     2
.  3:                     1     4     5     3
.  4:                  1     5     9     8     3
.  5:               1     6    14    17    11     4
.  6:            1     7    20    31    28    15     4
.  7:         1     8    27    51    59    43    19     5
.  8:      1     9    35    78   110   102    62    24     5
.  9:   1    10    44   113   188   212   164    86    29     6.
		

Links

• Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A035317, A007318, A074909.

Programs

Haskell

a181971 n k = a181971_tabl !! n !! k
a181971_row n = a181971_tabl !! n
a181971_tabl = map snd $ iterate f (1, [1]) where
   f (i, row) = (1 - i, zipWith (+) ([0] ++ row) (row ++ [i]))

Mathematica

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]]]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

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))))}
for(n=0,12,for(k=0,n,print1(T(n,k),","));print("")) \\ Paul D. Hanna, Jul 18 2012