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

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

Offset: 1

Author

Gary W. Adamson, Sep 10 2008

Keywords

• tabl
• sign

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.

Examples

			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;
  ...
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

Cf. A000225, A009545.

Programs

PARI

a25(n) = if (n, 2^n-1, 1); \\ A000225
a45(n) = (1+I)^(n-2) + (1-I)^(n-2); \\ A009545
T(n,k) = if (n>=k, a45(n-k+1)*a25(k-1), 0);
row(n) = vector(n, k, a45(n-k+1)*a25(k-1)); \\ Michel Marcus, Nov 20 2022

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.

This triangle = A*B.