A135091 - OEIS

A135091

A007318 * triangle M, where M = A002426 * 0^(n-k), 0<=k<=n.

1, 1, 1, 1, 2, 3, 1, 3, 9, 7, 1, 4, 18, 28, 19, 1, 5, 30, 70, 95, 51, 1, 6, 45, 140, 285, 306, 141, 1, 7, 63, 245, 665, 1071, 987, 393, 1, 8, 84, 392, 1330, 2856, 3948, 3144, 1107, 1, 9, 108, 588, 2394, 6426, 11844, 14148, 9963, 3139

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OFFSET

0,5

COMMENTS

Right border = A002426.

Row sums = A000984: (1, 2, 6, 20, 70, 252, ...).

The n-th row of this triangle lists the coefficients of the polynomial: p := (1/Pi)*Integral_{s=0..Pi} (1 + t - 2*t*cos(s))^n; Pi / 1 | n p := ---- | (1 + t - 2 t cos(s)) ds Pi | / 0 for example n=5 then 4 2 3 p = 19 t + 18 t + 28 t + 4 t + 1. - Theodore Kolokolnikov, Oct 09 2010

LINKS

Table of n, a(n) for n=0..54.

FORMULA

A007318 * triangle M, where M = A002426 * 0^(n-k), 0 <= k <= n; i.e., M = an infinite lower triangular matrix with A002426 as the right border and the rest zeros.

O.g.f. appears to be (1/sqrt(1-t*(1-x)))*1/sqrt(1-t*(1+3*x)) = 1 + (1+x)*t + (1 + 2*x + 3*x^2)*t^2 + ....

See A098473.

EXAMPLE

First few rows of the triangle:

1, 1;

1, 2, 3;

1, 3, 9, 7;

1, 4, 18, 28, 19;

1, 5, 30, 70, 95, 51;

1, 6, 45, 140, 285, 306, 141;

1, 7, 63, 245, 665, 1071, 987, 393;

...