1,2

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,...).

Left border = A009545, = expansion of sin(x)*exp(x) starting with offset 1.

Sum of row n terms = rightmost term of next row.

T(n,k) = A009545(n-k+1)*A000225(k-1).

A009545 = expansion of sin(x)*exp(x), starting with offset 1: (1, 2, 2, 0, -4, -8, -8,...).

A000225(k-1) = A000225 offset: (1, 1, 3, 7, 15, 31, 63, 127,...).

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.

Triangle A144081 = A*B.

First few rows of the triangle =

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;

0, -8, -24, -28, 0, 62, 126, 127;

16, 0, -24, -56, -60, 0, 126, 254, 255;

... 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.

A009545, Cf. A000225

Sequence in context: A089254 A140085 A071445 this_sequence A140086 A037194 A111630

Adjacent sequences: A144078 A144079 A144080 this_sequence A144082 A144083 A144084

tabl,sign

Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 10 2008