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A144081 Eigentriangle generated from expansion of sin(x)*exp(x), row sums = (2^n - 1). 3
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, 32, 16, 0, -56, -120, -124, 0, 254, 510, 511, 32, 32, 48, 0, -120, -248, -252, 0, 510, 1022, 1023 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..66.

FORMULA

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 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.

Triangle A144081 = A*B.

EXAMPLE

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.

CROSSREFS

A009545, Cf. A000225

Sequence in context: A279861 A140085 A071445 * A278838 A140086 A037194

Adjacent sequences:  A144078 A144079 A144080 * A144082 A144083 A144084

KEYWORD

tabl,sign

AUTHOR

Gary W. Adamson, Sep 10 2008

STATUS

approved

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Last modified June 22 23:15 EDT 2017. Contains 288633 sequences.