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%I #27 Jan 27 2023 12:12:05
%S 1,1,1,1,1,1,1,3,1,1,1,1,7,7,14,7,7,1,1,1,1,15,35,140,273,553,715,870,
%T 715,553,273,140,35,15,1,1,1,1,31,155,1240,6293,28861,105183,330460,
%U 876525,2020239,4032015,7063784,10855425,14743445,17678835,18796230
%N Triangle of numbers of inequivalent Boolean functions of n variables with exactly k nonzero values (atoms) under action of complementing group.
%D M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 143.
%H G. C. Greubel, <a href="/A054724/b054724.txt">Rows n = 0..10, flattened</a>
%H <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>
%F T(n,k) = 2^(-n)*C(2^n, k) if k is odd and 2^(-n)*(C(2^n, k) + (2^n-1)*C(2^(n-1), k/2)) if k is even.
%e Triangle begins:
%e k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 sums
%e n
%e 0 1 1 2
%e 1 1 1 1 3
%e 2 1 1 3 1 1 7
%e 3 1 1 7 7 14 7 7 1 1 46
%e 4 1 1 15 35 140 273 553 715 870 715 553 273 140 35 15 1 1 4336
%e ...
%p T:= (n, k)-> (binomial(2^n, k)+`if`(k::odd, 0,
%p (2^n-1)*binomial(2^(n-1), k/2)))/2^n:
%p seq(seq(T(n, k), k=0..2^n), n=0..5); # _Alois P. Heinz_, Jan 27 2023
%t rows = 5; t[n_, k_?OddQ] := 2^-n*Binomial[2^n, k]; t[n_, k_?EvenQ] := 2^-n*(Binomial[2^n, k] + (2^n-1)*Binomial[2^(n-1), k/2]); Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 0, 2^n}]] (* _Jean-François Alcover_, Nov 21 2011, after _Vladeta Jovovic_ *)
%t T[n_, k_]:= If[OddQ[k], Binomial[2^n, k]/2^n, 2^(-n)*(Binomial[2^n, k] + (2^n - 1)*Binomial[2^(n - 1), k/2])]; Table[T[n,k], {n,1,5}, {k,0,2^n}] //Flatten (* _G. C. Greubel_, Feb 15 2018 *)
%Y Row sums give A000231. Cf. A052265.
%K easy,nonn,nice,tabf
%O 0,8
%A _Vladeta Jovovic_, Apr 20 2000
%E Two terms for row n=0 prepended by _Alois P. Heinz_, Jan 27 2023
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