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A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

%I A191687 #14 Oct 20 2024 04:06:49
%S A191687 1,1,1,1,2,2,1,1,4,5,2,1,1,8,14,8,3,1,1,16,41,32,13,3,1,1,32,122,128,
%T A191687 63,18,4,1,1,64,365,512,313,108,25,4,1,1,128,1094,2048,1563,648,172,
%U A191687 32,5,1
%N A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.
%C A191687 T(n,k) is the number of compositions of even natural numbers into n parts <= k.
%H A191687 Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>
%e A191687 Top left corner:
%e A191687   1, 1, 1,  1,  1,...
%e A191687   1, 1, 2,  2,  3,...
%e A191687   1, 2, 5,  8, 13,...
%e A191687   1, 4,14, 32, 63,...
%e A191687   1, 8,41,128,313,...
%e A191687 T(2,4)=13: there are 13 compositions of even natural numbers into 2 parts <=4
%e A191687 0: (0,0);
%e A191687 2: (0,2), (2,0), (1,1);
%e A191687 4: (0,4), (4,0), (1,3), (3,1), (2,2);
%e A191687 6: (2,4), (4,2), (3,3);
%e A191687 8: (4,4).
%t A191687 Table[Table[Ceiling[1/2*((k+1)^n+(1+(-1)^k)/2)],{n,0,9},{k,0,9}]]
%Y A191687 Rows sums gives: A000012, A004526, A000982, A036486, A171714, A191484, A191489, A191494, A191495, A191496
%Y A191687 Columns sums gives: A000012, A000079, A007051, A004171, A034478, A081341, A034494, A092811, A083884, A093143
%K A191687 nonn,tabl
%O A191687 1,5
%A A191687 _Adi Dani_, Jun 11 2011