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A120903 Triangle T, read by rows, that satisfies matrix equation: T + (T-I)^2 = C, where C is Pascal's triangle.

%I A120903 #9 Jan 03 2021 15:31:02
%S A120903 1,1,1,-1,2,1,7,-3,3,1,-61,28,-6,4,1,751,-305,70,-10,5,1,-11821,4506,
%T A120903 -915,140,-15,6,1,226927,-82747,15771,-2135,245,-21,7,1,-5142061,
%U A120903 1815416,-330988,42056,-4270,392,-28,8,1,134341711,-46278549,8169372,-992964,94626,-7686,588,-36,9,1,-3975839341,1343417110
%N A120903 Triangle T, read by rows, that satisfies matrix equation: T + (T-I)^2 = C, where C is Pascal's triangle.
%C A120903 Column 0 is signed A048287, which is the number of semiorders on n labeled nodes.
%F A120903 E.g.f. A(x,y) satisfies: A(x,y) + [A(x,y) - exp(x*y)]^2 = exp(x+x*y).
%F A120903 Explicitly, e.g.f.: A(x,y) = exp(x*y)*(1 + sqrt(4*exp(x)-3))/2.
%F A120903 E.g.f. of column 0: (1 + sqrt(4*exp(x)-3))/2.
%F A120903 T(n,k) = -(-1)^(n-k)*A048287(n-k)*C(n,k) + 2*0^(n-k).
%F A120903 Matrix square: [T^2](n,k) = ( C(n,k) + 2*T(n,k) - 0^(n-k) )/2.
%e A120903 Triangle T begins:
%e A120903          1;
%e A120903          1,       1;
%e A120903         -1,       2,       1;
%e A120903          7,      -3,       3,     1;
%e A120903        -61,      28,      -6,     4,     1;
%e A120903        751,    -305,      70,   -10,     5,   1;
%e A120903     -11821,    4506,    -915,   140,   -15,   6,   1;
%e A120903     226927,  -82747,   15771, -2135,   245, -21,   7, 1;
%e A120903   -5142061, 1815416, -330988, 42056, -4270, 392, -28, 8, 1;
%e A120903 The matrix square of T less the diagonal is (T-I)^2:
%e A120903       0;
%e A120903       0,     0;
%e A120903       2,     0,   0;
%e A120903      -6,     6,   0,    0;
%e A120903      62,   -24,  12,    0,  0;
%e A120903    -750,   310, -60,   20,  0, 0;
%e A120903   11822, -4500, 930, -120, 30, 0, 0;
%e A120903 where C = T + (T-I)^2 = 2*T^2 - 2*T + I.
%o A120903 (PARI) /* Generated by Recursion T = C - (T-I)^2 : */ {T(n, k)=local(C=matrix(n+1, n+1, r, c, if(r>=c, binomial(r-1, c-1))), M=C); for(i=1, n+1, M=C-(M-M^0)^2 ); return(M[n+1, k+1])}
%o A120903 (PARI) /* Generated by E.G.F.: */ {T(n,k)=n!*polcoeff(polcoeff(exp(x*y)*(1 + sqrt(4*exp(x +x*O(x^n))-3))/2,n,x),k,y)}
%Y A120903 Cf. A048287 (column 0); A117269 (variant: T - (T-I)^2 = C).
%K A120903 sign,tabl
%O A120903 0,5
%A A120903 _Paul D. Hanna_, Jul 17 2006