A288870 Triangle T from array A(k,n) = (2*k+1)*2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.
2, 3, 4, 5, 7, 6, 9, 13, 11, 8, 17, 25, 21, 15, 10, 33, 49, 41, 29, 19, 12, 65, 97, 81, 57, 37, 23, 14, 129, 193, 161, 113, 73, 45, 27, 16, 257, 385, 321, 225, 145, 89, 53, 31, 18, 513, 769, 641, 449, 289, 177, 105, 61, 35, 20, 1025, 1537, 1281, 897, 577, 353, 209, 121, 69, 39, 22
Offset: 0
Examples
The array A begins: k\n 0 1 2 3 4 5 6 7 8 9 10 ... 0: 2 3 5 9 17 33 65 129 257 513 1025 1: 4 7 13 25 49 97 193 385 769 1537 3073 2: 6 11 21 41 81 161 321 641 1281 2561 5121 3: 8 15 29 57 113 225 449 897 1793 3585 7169 4: 10 19 37 73 145 289 577 1153 2305 4609 9217 5: 12 23 45 89 177 353 705 1409 2817 5633 11265 6: 14 27 53 105 209 417 833 1665 3329 6657 13313 7: 16 31 61 121 241 481 961 1921 3841 7681 15361 8: 18 35 69 137 273 545 1089 2177 4353 8705 17409 9: 20 39 77 153 305 609 1217 2433 4865 9729 19457 ... The triangle T begins: m\k 0 1 2 3 4 5 6 7 8 9 10 ... 0: 2 1: 3 4 2: 5 7 6 3: 9 13 11 8 4: 17 25 21 15 10 5: 33 49 41 29 19 12 6: 65 97 81 57 37 23 14 7: 129 193 161 113 73 45 27 16 8: 257 385 321 225 145 89 53 31 18 9: 513 769 641 449 289 177 105 61 35 20 10: 1025 1537 1281 897 577 353 209 121 69 39 22 ...
Crossrefs
Programs
-
Mathematica
Table[(2 k + 1)*2^(m - k) + 1, {m, 0, 10}, {k, 0, m}] // Flatten (* Michael De Vlieger, Jun 25 2017 *) -
PARI
A(n, k) = (2*n + 1)*2^k + 1; for(n=0, 10, for(k=0, n, print1(A(k, n - k),", "))) \\ Indranil Ghosh, Jun 22 2017
Formula
Array A(k, n) = (2*k+1)*2^n + 1 for k >= 0 and n >= 0.
Triangle T(m, k) = A(k, m-k) = (2*k+1)*2^(m-k) + 1, k >= m >= 0, otherwise T(m, k) = 0.
O.g.f. for column k of T: x^k*(2*(k+1) - (2*k+3)*x)/((1-2*x)*(1-x)), k >= 0.
E.g.f. for column k of T (without leading 0's): (2*k+1)*exp(2*x) + exp(x), k>=0.
E.g.f. for column k of T: 2^(-k)*(2*k+1)*exp(2*x) + exp(x) - S(k,x), with S(k, x) = 2^(-k)* Sum_{m=1..k} A288871(k,m)*x^(m-1)/(m-1)! if k >=1 and S(0,x) = 0.
Comments