A249111 Triangle of partial sums of rows in triangle A249095.

1, 1, 2, 3, 1, 2, 4, 5, 6, 1, 2, 5, 7, 10, 11, 12, 1, 2, 6, 9, 15, 18, 22, 23, 24, 1, 2, 7, 11, 21, 27, 37, 41, 46, 47, 48, 1, 2, 8, 13, 28, 38, 58, 68, 83, 88, 94, 95, 96, 1, 2, 9, 15, 36, 51, 86, 106, 141, 156, 177, 183, 190, 191, 192, 1, 2, 10, 17, 45, 66

Offset: 0

Author

Reinhard Zumkeller, Nov 14 2014

Keywords

• nonn
• tabf

Comments

Length of row n = 2*n+1.

In the layout as given in the example, T(n,k) is the sum of the two elements to the left and to the right of the element just above, with the row continued to the left by 0's and to the right by the last element, cf. formula. - M. F. Hasler, Nov 17 2014

Examples

			The triangle begins:
.  0:                            1
.  1:                        1   2   3
.  2:                    1   2   4   5   6
.  3:                1   2   5   7  10  11  12
.  4:             1  2   6   9  15  18  22  23  24
.  5:          1  2  7  11  21  27  37  41  46  47  48
.  6:       1  2  8 13  28  38  58  68  83  88  94  95  96
.  7:    1  2  9 15 36  51  86 106 141 156 177 183 190 191 192
.  8:  1 2 10 17 45 66 122 157 227 262 318 339 367 374 382 383 384 .
It can be seen that the elements (except for row 1) are sum of the neighbors to the upper left and upper right, with the table continued to the left with 0's and to the right with the last = largest element of each row. E.g., 1=0+1, 2=0+2, 4=1+3, 5=2+3 (=1+4 in the next row), 6=3+3 (in row 2), 7=2+5 etc. - _M. F. Hasler_, Nov 17 2014
		

Links

• Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A005408 (row lengths), A128543 (row sums), A248574 (central terms), A008949.

Programs

Haskell

a249111 n k = a249111_tabf !! n !! k
a249111_row n = a249111_tabf !! n
a249111_tabf = map (scanl1 (+)) a249095_tabf

PARI

T(n,k)=if(k<2,k+1,if(k>=2*n-2,3<<(n-1),T(n-1,k-2)+T(n-1,k))) \\ M. F. Hasler, Nov 17 2014

Formula

T(n,0) = A249095(n,0) = 1; T(n,k) = T(n,k-1) + A249095(n,k), k <= n.

T(n+1,k+1) = T(n,k-1) + T(n,k+1), with T(n,k-1)=0 for k<1 and T(n,k+1)=T(n,k) for k>=2n (last element of the row). In particular, T(n,k)=k+1 if k<2n and T(n,k)=3*2^(n-1) if k>=2n. - M. F. Hasler, Nov 17 2014