A115524 Number triangle (1,-x)+(x,x)/2+(x,-x)/2-(x^2,x^2) (expressed using the notation of stretched Riordan arrays).

1, 1, -1, -1, 0, 1, 0, 0, 1, -1, 0, -1, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Author

Paul Barry, Jan 25 2006

Keywords

• easy
• sign
• tabl

Examples

			Triangle begins
1,
1, -1,
-1, 0, 1,
0, 0, 1, -1,
0, -1, 0, 0, 1,
0, 0, 0, 0, 1, -1,
0, 0, -1, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1, -1,
0, 0, 0, -1, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 1, -1,
0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1,
0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1,
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of the triangle

Crossrefs

Row sums are A000007. Diagonal sums are A115525. Matrix inverse is A115526. Row sums of inverse are A023416(n+2).

Programs

PARI

up_to = 22154; \\ ; the first 210 rows of the 0-based triangle
A115524tr0(n,k) = ((-1)^n)*if(n==k+1, -(1+(-1)^k)/2, if(n==2*k+2, -1, (n==k)));
A115524list(up_to) = { my(v = vector(1+up_to), i=0); for(n=0,oo, for(k=0,n, i++; if(i > 1+up_to, return(v)); v[i] = A115524tr0(n,k))); (v); };
v115524 = A115524list(up_to);
A115524(n) = v115524[1+n]; \\ Antti Karttunen, Jan 20 2025

Formula

Column k has g.f. (-x)^k+(x(-x)^k+x^(k+1))/2-x^(2k+2).

Number triangle T(n, k)=(-1)^n*(if(n=k, 1, 0) OR if(n=2k+2, -1, 0) OR if(n=k+1, -(1+(-1)^k)/2, 0)).

G.f.: (1+x-x*y)/(1-x^2*y^2)-x^2/(1-x^2*y); - Paul Barry, Feb 02 2006