A158474 Triangle read by rows generated from (x-1)*(x-2)*(x-4)*...

1, 1, -1, 1, -3, 2, 1, -7, 14, -8, 1, -15, 70, -120, 64, 1, -31, 310, -1240, 1984, -1024, 1, -63, 1302, -11160, 41664, -64512, 32768, 1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152, 1, -255, 21590, -777240, 12850368, -99486720, 353730560

Offset: 0

Author

Gary W. Adamson, Mar 20 2009

Keywords

• tabl
• sign

Comments

Row sum of the unsigned triangle = A028361: (1, 2, 6, 30, 270, 4590, ...).

Right border of the unsigned triangle = A006125: (1, 1, 2, 8, 64, 1024, ...).

From Philippe Deléham, Mar 20 2009: (Start)

Unsigned triangle: A077957(n) DELTA A007179(n+1) = [1,0,2,0,4,0,8,0,16,0,32,0,...]DELTA[1,1,4,6,16,28,64,120,256,496,...], where DELTA is the operator defined in A084938.

Signed triangle: [1,0,2,0,4,0,8,0,16,0,...]DELTA[-1,-1,-4,-6,-16,-28,-64,...]. (End)

Examples

			First few rows of the triangle =
1;
1,  -1;
1,  -3,     2;
1,  -7,    14,     -8;
1, -15,    70,   -120,       64;
1, -31,   310,  -1240,     1984,    -1024;
1, -63,  1302, -11160,    41664,   -64512,     32768;
1,-127,  5334, -94488,   755904, -2731008,   4161536,  -2097152;
1,-255, 21590,-777240, 12850368,-99486720, 353730560,-534773760, 268435456;
...
Example: row 3 = x^3 - 7x^2 + 14x - 8 = (x-1)*(x-2)*(x-4).
		

Crossrefs

Cf. A028361, A006125.

Cf. A157963, A135950. - R. J. Mathar, Mar 20 2009

Programs

Maple

A158474 := proc(n,k) mul(x-2^j,j=0..n-1) ; expand(%); coeftayl(%,x=0,n-k) ; end proc: # R. J. Mathar, Aug 27 2011

Mathematica

{{1}}~Join~Table[Reverse@ CoefficientList[Fold[#1 (x - #2) &, 1, 2^Range[0, n]], x], {n, 0, 7}] // Flatten (* Michael De Vlieger, Dec 22 2016 *)

Formula

T(n,k) = coefficient [x^(n-k)] of (x-1)*(x-2)*(x-4)*...*(x-2^(n-1)).

T(n,k) = (-1)^k*(Sum_{j=0..k} T(k,j)*2^((k-j)*n))/(Product_{i=1..k} (2^i-1)) for n >= 0 and k > 0, i.e., e.g.f. of col k > 0 is: (-1)^k*(Sum_{j=0..k} T(k,j)* exp(2^(k-j)*t))/(Product_{i=1..k} (2^i-1)). - Werner Schulte, Dec 18 2016

T(n,k)/T(k,k) = A022166(n,k) for 0 <= k <= n. - Werner Schulte, Dec 21 2016