A192396 Square array T(n, k) = floor(((k+1)^n - (1+(-1)^k)/2)/2) read by antidiagonals.

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, 0, 128, 1093, 2048, 1562, 648, 171, 32, 4, 0, 0, 256, 3280, 8192, 7812, 3888, 1200, 256, 40, 5, 0

Offset: 0

Author

Adi Dani, Jun 29 2011

Keywords

• nonn
• tabl
• easy

Comments

T(n,k) is the number of compositions of odd natural numbers into n parts <=k.

Examples

			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;
    .        .      .       .    ...
with columns: A000004, A000079, A003462, A004171, A128531, A081341, ... .
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;
		

Links

• G. C. Greubel, Antidiagonals n = 0..50, flattened
• Adi Dani, Restricted compositions of natural numbers

Crossrefs

Rows: A000004, A004526, A007590, A036487, A191902, A191902.

Columns: A000004, A000079, A003462, A004171, A081341, A128531.

Programs

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

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

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