A133135 Third column of the inverse of the triangle of polynomial coefficients P(0,x)=1, 2P(n,x)=(1+x)*[(1+x)^(n-1)+x^(n-1)].

1, -2, 2, -1, 1, -4, 4, 13, -13, -142, 142, 1931, -1931, -36296, 36296, 893273, -893273, -27927346, 27927346, 1081725559, -1081725559, -50861556172, 50861556172, 2854289486309, -2854289486309, -188475382997654, 188475382997654, 14467150771771043, -14467150771771043

Offset: 0

Author

Paul Curtz, Sep 21 2007

Keywords

• sign

Comments

The triangle with [x^k] P(n,x) starts

1;

1, 1;

1/2, 3/2, 1;

1/2, 3/2, 2, 1;

1/2, 2, 3, 5/2, 1;

1/2, 5/2, 5, 5, 3, 1;

1/2, 3,15/2, 10,15/2, 7/2, 1;

1/2, 7/2,21/2,35/2,35/2,21/2, 4, 1;

1/2, 4, 14, 28, 35, 28, 14, 9/2, 1;

1/2, 9/2, 18, 42, 63, 63, 42, 18, 5, 1;

The sum of the rows of this triangle is A094373 (previously noticed by Paul Curtz). - Jean-François Alcover, Jul 22 2013

Apparently a(2*n) = A102055(n) and a(2*n+1) = -a(2*n) for n >= 0. - Georg Fischer, Dec 05 2022

Examples

			The inverse of the triangle of coefficients starts
1;
-1, 1;
1, -3/2, 1;
-1, 3/2, -2, 1;
1, -5/4, 2, -5/2, 1;
-1, 5/4 -1, 5/2, -3, 1;
1, -7/4, 1, 0, 3, -7/2, 1;
-1, 7/4,-4, 0, 2, 7/2,-4, 1;
1, 3/8, 4,-21/2, -2, 21/4, 4,-9/2, 1;
-1, -3/8,13, 21/2,-26,-21/4,10, 9/2,-5, 1;
and defines the sequence in its third column.
Apart from the numbers along the diagonal, the absolute values show up in pairs if read along columns. Conjectures: Starting with the third line, columns are alternatingly fractions and integers. The row sums (1, 0, 1/2, -1/2, 1/4, -1/4, 3/4, -3/4, -11/8, 11/8,..) also show up in pairs from the third row on.
		

Programs

Mathematica

max = 28; p[0, ] = 1; p[n, x_] := (1 + x)*((1 + x)^(n - 1) + x^(n - 1))/2; t = Table[ Coefficient[p[n, x], x, k], {n, 0, max + 2}, {k, 0, max + 2}]; a[n_] := Inverse[t][[All, 3]][[n + 3]]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jul 22 2013 *)

Extensions

Edited and extended by R. J. Mathar, Aug 02 2008