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 A158474 Triangle read by rows generated from (x-1)*(x-2)*(x-4)*... 6
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 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) LINKS 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 EXAMPLE 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). 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 *) CROSSREFS Cf. A028361, A006125. Cf. A157963, A135950. - R. J. Mathar, Mar 20 2009 Sequence in context: A082038 A143774 A196842 * A090452 A305538 A193924 Adjacent sequences:  A158471 A158472 A158473 * A158475 A158476 A158477 KEYWORD tabl,sign AUTHOR Gary W. Adamson, Mar 20 2009 STATUS approved

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Last modified July 31 10:51 EDT 2021. Contains 346373 sequences. (Running on oeis4.)