Search: id:A144081
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%I A144081
%S A144081 1,2,1,2,2,3,0,2,6,7,4,0,6,14,15,8,4,0,14,30,31,8,8,12,0,30,62,63,
%T A144081 0,8,24,28,0,62,126,127,16,0,24,56,60,0,126,254,255,32,16,0,56,
%U A144081 120,124,0,254,510,511,32,32,48,0,120,248,252,0,510,1022,1023
%V A144081 1,2,1,2,2,3,0,2,6,7,-4,0,6,14,15,-8,-4,0,14,30,31,-8,-8,-12,0,30,62,63,
%W A144081 0,-8,-24,-28,0,62,126,127,16,0,-24,-56,-60,0,126,254,255,32,16,0,-56,
%X A144081 -120,-124,0,254,510,511,32,32,48,0,-120,-248,-252,0,510,1022,1023
%N A144081 Eigentriangle generated from expansion of sin(x)*exp(x), row sums = (2^n 
               - 1).
%C A144081 Row sums = (2^n - 1): (1, 3, 7, 15, 31,...) = INVERT transform of A009545
%C A144081 starting with offset 1. Right border = (1, 1, 3, 7, 15,...).
%C A144081 Left border = A009545, = expansion of sin(x)*exp(x) starting with offset 
               1.
%C A144081 Sum of row n terms = rightmost term of next row.
%F A144081 T(n,k) = A009545(n-k+1)*A000225(k-1).
%F A144081 A009545 = expansion of sin(x)*exp(x), starting with offset 1: (1, 2, 
               2, 0, -4, -8, -8,...).
%F A144081 A000225(k-1) = A000225 offset: (1, 1, 3, 7, 15, 31, 63, 127,...).
%F A144081 These opertions = the following: Matrix A = an infinite lower triangular 
               matrix with rows = A009545 subsequences decrescendo: (1; 2,1; 2,2,
               1; 0,2,2,1; -4,0,2,2,1;...) and matrix B = an infinite lower triangular 
               matrix with (1, 1, 3, 7, 15,...) in the main diagonal and the rest 
               zeros.
%F A144081 Triangle A144081 = A*B.
%e A144081 First few rows of the triangle =
%e A144081 1;
%e A144081 2, 1;
%e A144081 2, 2, 3;
%e A144081 0, 2, 6, 7;
%e A144081 -4, 0, 6, 14, 15;
%e A144081 -8, -4, 0, 14, 30, 31;
%e A144081 -8, -8, -12, 0, 30, 62, 63;
%e A144081 0, -8, -24, -28, 0, 62, 126, 127;
%e A144081 16, 0, -24, -56, -60, 0, 126, 254, 255;
%e A144081 ... Example: row 4 = (0, 2, 6, 7) pairwise product of (0, 2, 2, 1) and 
               (1, 1, 3, 7) = (0*1, 2*1, 2*3, 1*7); where (1, 2, 2, 0,...) = the 
               first 4 terms of A009545 with offset 1.
%Y A144081 A009545, Cf. A000225
%Y A144081 Sequence in context: A089254 A140085 A071445 this_sequence A140086 A037194 
               A111630
%Y A144081 Adjacent sequences: A144078 A144079 A144080 this_sequence A144082 A144083 
               A144084
%K A144081 tabl,sign
%O A144081 1,2
%A A144081 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 10 2008

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