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

%I A192396 #24 Jul 11 2023 09:17:28
%S A192396 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,
%T A192396 121,128,62,18,3,0,0,64,364,512,312,108,24,4,0,0,128,1093,2048,1562,
%U A192396 648,171,32,4,0,0,256,3280,8192,7812,3888,1200,256,40,5,0
%N A192396 Square array T(n, k) = floor(((k+1)^n - (1+(-1)^k)/2)/2) read by antidiagonals.
%C A192396 T(n,k) is the number of compositions of odd natural numbers into n parts <=k.
%H A192396 G. C. Greubel, <a href="/A192396/b192396.txt">Antidiagonals n = 0..50, flattened</a>
%H A192396 Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>
%e A192396 T(2,4)=12: there are 12 compositions of odd natural numbers into 2 parts <=4
%e A192396   1: (0,1), (1,0);
%e A192396   3: (1,2), (2,1), (0,3), (3,0);
%e A192396   5: (1,4), (4,1), (2,3), (3,2);
%e A192396   7: (3,4), (4,3).
%e A192396 The table starts
%e A192396     0,  0,   0,   0,    0,    0, ... A000004;
%e A192396     0,  1,   1,   2,    2,    3, ... A004526;
%e A192396     0,  2,   4,   8,   12,   18, ... A007590;
%e A192396     0,  4,  13,  32,   62,  108, ... A036487;
%e A192396     0,  8,  40, 128,  312,  648, ... A191903;
%e A192396     0, 16, 121, 512, 1562, 3888, ... A191902;
%e A192396     .        .      .       .    ...
%e A192396 with columns: A000004, A000079, A003462, A004171, A128531, A081341, ... .
%e A192396 Antidiagonal triangle begins:
%e A192396   0;
%e A192396   0,  0;
%e A192396   0,  1,   0;
%e A192396   0,  2,   1,   0;
%e A192396   0,  4,   4,   2,   0;
%e A192396   0,  8,  13,   8,   2,   0;
%e A192396   0, 16,  40,  32,  12,   3,  0;
%e A192396   0, 32, 121, 128,  62,  18,  3,  0;
%e A192396   0, 64, 364, 512, 312, 108, 24,  4,  0;
%p A192396 A192396 := proc(n,k) (k+1)^n-(1+(-1)^k)/2 ; floor(%/2) ; end proc:
%p A192396 seq(seq( A192396(d-k,k),k=0..d),d=0..10) ; # _R. J. Mathar_, Jun 30 2011
%t A192396 T[n_, k_]:= Floor[((k+1)^n - (1+(-1)^k)/2)/2];
%t A192396 Table[T[n-k,k], {n,0,12}, {k,0,n}]//Flatten
%o A192396 (Magma)
%o A192396 A192396:= func< n,k | Floor(((k+1)^n - (1+(-1)^k)/2)/2) >;
%o A192396 [A192396(n-k,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 11 2023
%o A192396 (SageMath)
%o A192396 def A192396(n,k): return ((k+1)^n - ((k+1)%2))//2
%o A192396 flatten([[A192396(n-k,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 11 2023
%Y A192396 Rows: A000004, A004526, A007590, A036487, A191902, A191902.
%Y A192396 Columns: A000004, A000079, A003462, A004171, A081341, A128531.
%K A192396 nonn,tabl,easy
%O A192396 0,8
%A A192396 _Adi Dani_, Jun 29 2011