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A288870 Triangle T from array A(k,n) = (2*k+1)*2^n + 1, k >=0, n >= 0 read by downwards antidiagonals. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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: A283192 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

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Last modified September 22 17:39 EDT 2021. Contains 347607 sequences. (Running on oeis4.)