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
Examples
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.
Programs
-
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)}
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.
Comments