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

0,8

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

Adi Dani, Restricted compositions of natural numbers

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;
    .        .      .       .    ...
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;

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

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

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

Sequence in context: A089975 A034366 A121465 * A094449 A274776 A274777

Adjacent sequences: A192393 A192394 A192395 * A192397 A192398 A192399

Keyword

nonn,tabl,easy

Author

Adi Dani, Jun 29 2011

Status

approved