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

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, 134341711, -46278549, 8169372, -992964, 94626, -7686, 588, -36, 9, 1, -3975839341, 1343417110

Offset

0,5

Comments

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

Links

Table of n, a(n) for n=0..56.

Formula

E.g.f. A(x,y) satisfies: A(x,y) + [A(x,y) - exp(x*y)]^2 = exp(x+x*y).

Explicitly, 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.

Example

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.

Prog

(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)}

Crossrefs

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

Sequence in context: A125697 A090699 A214550 * A180335 A257699 A160535

Adjacent sequences: A120900 A120901 A120902 * A120904 A120905 A120906

Keyword

sign,tabl

Author

Paul D. Hanna, Jul 17 2006

Status

approved