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

%I A115524 #9 Jan 20 2025 10:22:08
%S A115524 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 A115524 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 A115524 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 A115524 Number triangle (1,-x)+(x,x)/2+(x,-x)/2-(x^2,x^2) (expressed using the notation of stretched Riordan arrays).
%H A115524 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 A115524 Column k has g.f. (-x)^k+(x(-x)^k+x^(k+1))/2-x^(2k+2).
%F A115524 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 A115524 G.f.: (1+x-x*y)/(1-x^2*y^2)-x^2/(1-x^2*y); - _Paul Barry_, Feb 02 2006
%e A115524 Triangle begins
%e A115524 1,
%e A115524 1, -1,
%e A115524 -1, 0, 1,
%e A115524 0, 0, 1, -1,
%e A115524 0, -1, 0, 0, 1,
%e A115524 0, 0, 0, 0, 1, -1,
%e A115524 0, 0, -1, 0, 0, 0, 1,
%e A115524 0, 0, 0, 0, 0, 0, 1, -1,
%e A115524 0, 0, 0, -1, 0, 0, 0, 0, 1,
%e A115524 0, 0, 0, 0, 0, 0, 0, 0, 1, -1,
%e A115524 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1,
%e A115524 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1,
%e A115524 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1,
%e A115524 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1,
%o A115524 (PARI)
%o A115524 up_to = 22154; \\ ; the first 210 rows of the 0-based triangle
%o A115524 A115524tr0(n,k) = ((-1)^n)*if(n==k+1, -(1+(-1)^k)/2, if(n==2*k+2, -1, (n==k)));
%o A115524 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 A115524 v115524 = A115524list(up_to);
%o A115524 A115524(n) = v115524[1+n]; \\ _Antti Karttunen_, Jan 20 2025
%Y A115524 Row sums are A000007. Diagonal sums are A115525. Matrix inverse is A115526. Row sums of inverse are A023416(n+2).
%K A115524 easy,sign,tabl
%O A115524 0,1
%A A115524 _Paul Barry_, Jan 25 2006