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A249111 Triangle of partial sums of rows in triangle A249095.

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