|
%I #11 Nov 19 2014 10:59:37
%S 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,
%T 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,
%U 51,86,106,141,156,177,183,190,191,192,1,2,10,17,45,66
%N Triangle of partial sums of rows in triangle A249095.
%C Length of row n = 2*n+1.
%C 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
%H Reinhard Zumkeller, <a href="/A249111/b249111.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,0) = A249095(n,0) = 1; T(n,k) = T(n,k-1) + A249095(n,k), k <= n.
%F 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
%e The triangle begins:
%e . 0: 1
%e . 1: 1 2 3
%e . 2: 1 2 4 5 6
%e . 3: 1 2 5 7 10 11 12
%e . 4: 1 2 6 9 15 18 22 23 24
%e . 5: 1 2 7 11 21 27 37 41 46 47 48
%e . 6: 1 2 8 13 28 38 58 68 83 88 94 95 96
%e . 7: 1 2 9 15 36 51 86 106 141 156 177 183 190 191 192
%e . 8: 1 2 10 17 45 66 122 157 227 262 318 339 367 374 382 383 384 .
%e 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
%o (Haskell)
%o a249111 n k = a249111_tabf !! n !! k
%o a249111_row n = a249111_tabf !! n
%o a249111_tabf = map (scanl1 (+)) a249095_tabf
%o (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
%Y Cf. A005408 (row lengths), A128543 (row sums), A248574 (central terms), A008949.
%K nonn,tabf
%O 0,3
%A _Reinhard Zumkeller_, Nov 14 2014
|
