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

0,1

Comments

This entry was motivated by a class work of Ferran D.

Links

Table of n, a(n) for n=0..65.

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.

Example

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
...

Mathematica

Table[(2 k + 1)*2^(m - k) + 1, {m, 0, 10}, {k, 0, m}] // Flatten (* Michael De Vlieger, Jun 25 2017 *)

Prog

(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

Crossrefs

Cf. A288871. Columns of T (no 0's, or rows of A): A000051, A181565, A083575, A083686, A083705, A083683, A168596.

Sequence in context: A369282 A266638 A256231 * A283194 A254498 A185969

Adjacent sequences: A288867 A288868 A288869 * A288871 A288872 A288873

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang, Jun 21 2017

Status

approved