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A144081 Eigentriangle generated from expansion of sin(x)*exp(x), row sums = (2^n - 1).

%I A144081 #10 Nov 20 2022 10:53:30
%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,
%T A144081 63,0,-8,-24,-28,0,62,126,127,16,0,-24,-56,-60,0,126,254,255,32,16,0,
%U A144081 -56,-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 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 operations = 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 This triangle = 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   ...
%e A144081 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.
%o A144081 (PARI) a25(n) = if (n, 2^n-1, 1); \\ A000225
%o A144081 a45(n) = (1+I)^(n-2) + (1-I)^(n-2); \\ A009545
%o A144081 T(n,k) = if (n>=k, a45(n-k+1)*a25(k-1), 0);
%o A144081 row(n) = vector(n, k, a45(n-k+1)*a25(k-1)); \\ _Michel Marcus_, Nov 20 2022
%Y A144081 Cf. A000225, A009545.
%K A144081 tabl,sign
%O A144081 1,2
%A A144081 _Gary W. Adamson_, Sep 10 2008