0,5

Column 0 is signed A048287, which is the number of semiorders on n labeled nodes.

E.g.f. A(x,y) satisfies: A(x,y) + [A(x,y) - exp(x*y)]^2 = exp(x+x*y); explicity, e.g.f: A(x,y) = exp(x*y)*(1 + sqrt(4*exp(x)-3))/2. E.g.f. of column 0: (1 + sqrt(4*exp(x)-3))/2. T(n,k) = -(-1)^(n-k)*A048287(n-k)*C(n,k) + 2*0^(n-k). Matrix square: [T^2](n,k) = ( C(n,k) + 2*T(n,k) - 0^(n-k) )/2.

Triangle T begins:

1;

1, 1;

-1, 2, 1;

7, -3, 3, 1;

-61, 28, -6, 4, 1;

751, -305, 70, -10, 5, 1;

-11821, 4506, -915, 140, -15, 6, 1;

226927, -82747, 15771, -2135, 245, -21, 7, 1;

-5142061, 1815416, -330988, 42056, -4270, 392, -28, 8, 1;

The matrix square of T less the diagonal is (T-I)^2:

0;

0, 0;

2, 0, 0;

-6, 6, 0, 0;

62, -24, 12, 0, 0;

-750, 310, -60, 20, 0, 0;

11822, -4500, 930, -120, 30, 0, 0;

where C = T + (T-I)^2 = 2*T^2 - 2*T + I.

(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])} (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)}

Cf. A048287 (column 0); A117269 (variant: T - (T-I)^2 = C).

Sequence in context: A091370 A125697 A090699 this_sequence A021050 A115629 A144696

Adjacent sequences: A120900 A120901 A120902 this_sequence A120904 A120905 A120906

sign,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 17 2006