|
| |
|
|
A131239
|
|
Triangle, T(n,k) = 3*A007318(n,k) - 2*A046854(n,k), read by rows.
|
|
2
|
|
|
|
1, 1, 1, 1, 4, 1, 1, 5, 7, 1, 1, 8, 12, 10, 1, 1, 9, 24, 22, 13, 1, 1, 12, 33, 52, 35, 16, 1, 1, 13, 51, 85, 95, 51, 19, 1, 1, 16, 64, 148, 180, 156, 70, 22, 1, 1, 17, 88, 212, 348, 336, 238, 92, 25, 1, 1, 20, 105, 320, 560, 714, 574, 344, 117, 28, 1, 1, 21, 135, 425, 920, 1274, 1330, 918, 477, 145, 31, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
Row sums = A074878: (1, 2, 6, 14, 32, 70, 239, ...).
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n,k) = 3*A007318(n,k) - 2*A046854(n,k) as infinite lower triangular matrices.
T(n,k) = 3*binomial(n,k) - 2*binomial(floor((n+k)/2), k). - G. C. Greubel, Jul 12 2019
|
|
|
EXAMPLE
|
First few rows of the triangle:
1;
1, 1;
1, 4, 1;
1, 5, 7, 1;
1, 8, 12, 10, 1;
1, 9, 24, 22, 13, 1;
1, 12, 33, 52, 35, 16, 1;
...
|
|
|
MATHEMATICA
|
With[{B=Binomial}, Table[3*B[n, k] - 2*B[Floor[(n+k)/2], k], {n, 0, 12}, {k, 0, n}]]//Flatten (* G. C. Greubel, Jul 12 2019 *)
|
|
|
PROG
|
(PARI) b=binomial; T(n, k) = 3*b(n, k) - 2*b((n+k)\2, k);
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 12 2019
(Magma) B:=Binomial; [3*B(n, k) - 2*B(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2019
(Sage) b=binomial; [[3*b(n, k) - 2*b(floor((n+k)/2), k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 12 2019
(GAP) B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> 3*B(n, k) - 2*B(Int((n+k)/2), k) ))); # G. C. Greubel, Jul 12 2019
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
