OFFSET
0,1
COMMENTS
This entry was motivated by a class work of Ferran D.
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
KEYWORD
AUTHOR
Wolfdieter Lang, Jun 21 2017
STATUS
approved