what is the circuit of 2^n -1

i am sorry my english is poor.

if it is just that simple you could write

If I understood the question (4 to 16 decoder) then MSchmitt's coding is how I would implement this as well. If you don't want the decoder registered then you would write it like this:

First could you clarify? Are you trying to get:

module pow_2toN(
    input          n,
    output        result
);
wire     shift_reg;
assign  shift_reg   = {16'd0,16'hffff} << n;
assign  result      = shift_reg;
endmodule

module pow_2toN(
    input          n,
    output        result
);
genvar i;
generate
    for(i=0;i<16;i=i+1) begin : result_assigns
        assign result = (i < n) : 1'b1 : 1'b0;
    end
endgenerate
endmodule

module pow_2toN(
    input          n,
    output  reg   result
);
integer i;
always @* begin
                    result      = 16'd0;
    for(i=0;i<16;i=i+1)
        if(i<n)     result   = 1'b1;
end
endmodule

module pow_2toN(
    input          n,
    output  reg   result
);
integer i;
always @* begin
    result      = 16'd1;
    for(i=1;i<16;i=i+1)
        if(i<=n)    result      = {result,1'b0};
                    result      = result + 16'hffff;
end
endmodule

Oh Jake great , thats real fun to read and think about.

As I stipulated, my assumption was that he wanted (2^n)-1. So the desired result would in fact be (0,1,3,7,f,...). If that assumption is incorrect then all of the examples need tweaking.

module pow_2toN(
    input          n,
    output        result
);
 
wire     shift_reg;
 
assign  shift_reg   = {16'd0,16'hffff} << n;
assign  result      = shift_reg;
 
endmodule

module pow_2toN(
    input          n,
    output        result
);
 
genvar i;
generate
    for(i=0;i<16;i=i+1) begin : result_assigns
        assign result = (i < n) : 1'b1 : 1'b0;
    end
endgenerate
 
endmodule

module pow_2toN(
    input          n,
    output  reg   result
);
 
integer i;
always @* begin
                    result      = 16'd0;
    for(i=0;i<16;i=i+1)
        if(i<n)     result   = 1'b1;
end
 
endmodule

module pow_2toN(
    input          n,
    output  reg   result
);
 
integer i;
always @* begin
    result      = 16'd1;
 
    for(i=1;i<16;i=i+1)
        if(i<=n)    result      = {result,1'b0};
 
                    result      = result + 16'hffff;
end
 
endmodule

Hi jakobjones

If you have access to synplify, try the various implementations and see what circuits they produce.