OFFSET
0,8
COMMENTS
T(n,k) is the number of compositions of odd natural numbers into n parts <=k.
LINKS
G. C. Greubel, Antidiagonals n = 0..50, flattened
EXAMPLE
T(2,4)=12: there are 12 compositions of odd natural numbers into 2 parts <=4
1: (0,1), (1,0);
3: (1,2), (2,1), (0,3), (3,0);
5: (1,4), (4,1), (2,3), (3,2);
7: (3,4), (4,3).
The table starts
0, 0, 0, 0, 0, 0, ... A000004;
0, 1, 1, 2, 2, 3, ... A004526;
0, 2, 4, 8, 12, 18, ... A007590;
0, 4, 13, 32, 62, 108, ... A036487;
0, 8, 40, 128, 312, 648, ... A191903;
0, 16, 121, 512, 1562, 3888, ... A191902;
. . . . ...
Antidiagonal triangle begins:
0;
0, 0;
0, 1, 0;
0, 2, 1, 0;
0, 4, 4, 2, 0;
0, 8, 13, 8, 2, 0;
0, 16, 40, 32, 12, 3, 0;
0, 32, 121, 128, 62, 18, 3, 0;
0, 64, 364, 512, 312, 108, 24, 4, 0;
MAPLE
A192396 := proc(n, k) (k+1)^n-(1+(-1)^k)/2 ; floor(%/2) ; end proc:
seq(seq( A192396(d-k, k), k=0..d), d=0..10) ; # R. J. Mathar, Jun 30 2011
MATHEMATICA
T[n_, k_]:= Floor[((k+1)^n - (1+(-1)^k)/2)/2];
Table[T[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Magma)
A192396:= func< n, k | Floor(((k+1)^n - (1+(-1)^k)/2)/2) >;
[A192396(n-k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 11 2023
(SageMath)
def A192396(n, k): return ((k+1)^n - ((k+1)%2))//2
flatten([[A192396(n-k, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 11 2023
CROSSREFS
KEYWORD
AUTHOR
Adi Dani, Jun 29 2011
STATUS
approved