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A115359 Matrix (1,x)-(x,x^2) in Riordan array notation.

%I A115359 #17 Jan 19 2025 14:16:14
%S A115359 1,-1,1,0,0,1,0,-1,0,1,0,0,0,0,1,0,0,-1,0,0,1,0,0,0,0,0,0,1,0,0,0,-1,
%T A115359 0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,
%U A115359 0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A115359 Matrix (1,x)-(x,x^2) in Riordan array notation.
%H A115359 Antti Karttunen, <a href="/A115359/b115359.txt">Table of n, a(n) for n = 0..22154; the first 210 rows of the triangle</a>
%F A115359 Number triangle T(n, k)=if(n=k, 1, 0) OR if(n=2k+1, -1, 0).
%F A115359 a(n) = A010054(n) - A379480(n). [As a flat sequence with starting offset 1] - _Antti Karttunen_, Jan 19 2025
%e A115359 Triangle begins:
%e A115359 n\k|  0   1   2   3   4   5   6   7   8   9
%e A115359 ---+-----------------------------------------
%e A115359 0  |  1;
%e A115359 1  | -1,  1;
%e A115359 2  |  0,  0,  1;
%e A115359 3  |  0, -1,  0,  1;
%e A115359 4  |  0,  0,  0,  0,  1;
%e A115359 5  |  0,  0, -1,  0,  0,  1;
%e A115359 6  |  0,  0,  0,  0,  0,  0,  1;
%e A115359 7  |  0,  0,  0, -1,  0,  0,  0,  1;
%e A115359 8  |  0,  0,  0,  0,  0,  0,  0,  0,  1;
%e A115359 9  |  0,  0,  0,  0, -1,  0,  0,  0,  0,  1;
%e A115359 etc. Row and column numbering added by _Antti Karttunen_, Jan 19 2025
%o A115359 (PARI) tabl(nn) = {T = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); for (n=1, nn, for (k=1, n, print1(T[n, k], ", ");); print(););} \\ _Michel Marcus_, Mar 28 2015
%o A115359 (PARI)
%o A115359 A115359off1(n) = (ispolygonal(n,3)-(!(n%2) && issquare(n/2))); \\ (This is one-based)
%o A115359 A115359(n) = A115359off1(1+n); \\ (zero-based) - _Antti Karttunen_, Jan 19 2025
%Y A115359 Row sums are 1,0,1,0,1,0... (A059841), Diagonal sums are A115360. Inverse is A115361.
%Y A115359 Cf. A000217, A001105, A010054, A379480.
%Y A115359 Cf. also A115356.
%K A115359 easy,sign,tabl
%O A115359 0,1
%A A115359 _Paul Barry_, Jan 21 2006