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

%I A158474 #27 Dec 24 2016 11:15:15
%S A158474 1,1,-1,1,-3,2,1,-7,14,-8,1,-15,70,-120,64,1,-31,310,-1240,1984,-1024,
%T A158474 1,-63,1302,-11160,41664,-64512,32768,1,-127,5334,-94488,755904,
%U A158474 -2731008,4161536,-2097152,1,-255,21590,-777240,12850368,-99486720,353730560
%N A158474 Triangle read by rows generated from (x-1)*(x-2)*(x-4)*...
%C A158474 Row sum of the unsigned triangle = A028361: (1, 2, 6, 30, 270, 4590, ...).
%C A158474 Right border of the unsigned triangle = A006125: (1, 1, 2, 8, 64, 1024, ...).
%C A158474 From _Philippe Deléham_, Mar 20 2009: (Start)
%C A158474 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.
%C A158474 Signed triangle: [1,0,2,0,4,0,8,0,16,0,...]DELTA[-1,-1,-4,-6,-16,-28,-64,...]. (End)
%F A158474 T(n,k) = coefficient [x^(n-k)] of (x-1)*(x-2)*(x-4)*...*(x-2^(n-1)).
%F A158474 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
%F A158474 T(n,k)/T(k,k) = A022166(n,k) for 0 <= k <= n. - _Werner Schulte_, Dec 21 2016
%e A158474 First few rows of the triangle =
%e A158474 1;
%e A158474 1,  -1;
%e A158474 1,  -3,     2;
%e A158474 1,  -7,    14,     -8;
%e A158474 1, -15,    70,   -120,       64;
%e A158474 1, -31,   310,  -1240,     1984,    -1024;
%e A158474 1, -63,  1302, -11160,    41664,   -64512,     32768;
%e A158474 1,-127,  5334, -94488,   755904, -2731008,   4161536,  -2097152;
%e A158474 1,-255, 21590,-777240, 12850368,-99486720, 353730560,-534773760, 268435456;
%e A158474 ...
%e A158474 Example: row 3 = x^3 - 7x^2 + 14x - 8 = (x-1)*(x-2)*(x-4).
%p A158474 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
%t A158474 {{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 *)
%Y A158474 Cf. A028361, A006125.
%Y A158474 Cf. A157963, A135950. - _R. J. Mathar_, Mar 20 2009
%K A158474 tabl,sign
%O A158474 0,5
%A A158474 _Gary W. Adamson_, Mar 20 2009