A022619 Triangle T(n,k)of numbers of asymmetric Boolean functions of n variables with exactly k = 0..2^n nonzero values (atoms) under action of complementing group C(n,2).

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 7, 7, 7, 0, 1, 0, 0, 1, 0, 35, 105, 273, 448, 715, 750, 715, 448, 273, 105, 35, 0, 1, 0, 0, 1, 0, 155, 1085, 6293, 27776, 105183, 327050, 876525, 2011776, 4032015, 7048811, 10855425, 14721280, 17678835, 18771864

Offset: 1

Author

Vladeta Jovovic, Jul 13 2000

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  [0,1,0],
  [0,1,0,1,0],
  [0,1,0,7,7,7,0,1,0],
  ...;
T(5,k) = coefficient of x^k in (1/32)*((1+x)^32-31*(1+x^2)^16+310*(1+x^4)^8-1240*(1+x^8)^4+1984*(1+x^16)^2-1024*(1+x^32)),k = 0..32.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 10 rows, flattened
• Index entries for sequences related to Boolean functions

Crossrefs

Row sums give A051502.

Cf. A054724.

Programs

Mathematica

T[n_,0]:=0; T[n_, k_] := (1/2^n)*Coefficient[Sum[(-1)^j*2^(Binomial[j, 2])* QBinomial[n, j, 2]*(1 + x^(2^j))^(2^(n - j)), {j, 0, n}], x^k];
Table[T[n, k], {n, 1, 5}, {k, 0, 2^n}] // Flatten (* G. C. Greubel, Feb 15 2018 *)

Formula

T(n, k) = coefficient of x^k in (1/2^n)*Sum_{j = 0..n} (-1)^j*2^C(j, 2)*[n, j]*(1+x^(2^j))^(2^(n-j)), where [n, j] is Gaussian 2-binomial coefficient; k = 0..2^n.