A115524 - OEIS

%I #9 Jan 20 2025 10:22:08

%S 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,

%T 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,

%U 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

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

%H Antti Karttunen, <a href="/A115524/b115524.txt">Table of n, a(n) for n = 0..22154; the first 210 rows of the triangle</a>

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

%F 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)).

%F G.f.: (1+x-x*y)/(1-x^2*y^2)-x^2/(1-x^2*y); - _Paul Barry_, Feb 02 2006

%e Triangle begins

%e 1,

%e 1, -1,

%e -1, 0, 1,

%e 0, 0, 1, -1,

%e 0, -1, 0, 0, 1,

%e 0, 0, 0, 0, 1, -1,

%e 0, 0, -1, 0, 0, 0, 1,

%e 0, 0, 0, 0, 0, 0, 1, -1,

%e 0, 0, 0, -1, 0, 0, 0, 0, 1,

%e 0, 0, 0, 0, 0, 0, 0, 0, 1, -1,

%e 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1,

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1,

%e 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1,

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1,

%o (PARI)

%o up_to = 22154; \\ ; the first 210 rows of the 0-based triangle

%o A115524tr0(n,k) = ((-1)^n)*if(n==k+1, -(1+(-1)^k)/2, if(n==2*k+2, -1, (n==k)));

%o 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); };

%o v115524 = A115524list(up_to);

%o A115524(n) = v115524[1+n]; \\ _Antti Karttunen_, Jan 20 2025

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

%K easy,sign,tabl

%O 0,1

%A _Paul Barry_, Jan 25 2006