A116466 -id:A116466

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A114700

Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I), where T(n,k) = [T^-1](n-1,k) + [T^-1](n-1,k-1) for n>k>0, with T(n,0)=T(n,n)=1 for n>=0 and I is the identity matrix.

+10
2

1, 1, 1, 1, 0, 1, 1, -1, 1, 1, 1, 0, 0, 0, 1, 1, -1, 0, 0, 1, 1, 1, 0, 1, 0, -1, 0, 1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 0, 2, 2, 0, -2, -2, 0, 1, 1, -1, -2, -4, -2, 2, 4, 2, 1, 1, 1, 0, 3, 6, 6, 0, -6, -6, -3, 0, 1, 1, -1, -3, -9, -12, -6, 6, 12, 9, 3, 1, 1, 1, 0, 4, 12, 21, 18, 0, -18, -21, -12, -4, 0, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,39

COMMENTS

The rows of this triangle are symmetric up to sign. Row sums = 2 after row 0. Unsigned row sums = A116466. Row squared sums = A116467. Central terms of odd rows: T(2*n+1,n+1) = |A064310(n)|.

LINKS

Table of n, a(n) for n=0..90.

FORMULA

G.f.: A(x,y) = 1/(1-x*y)+ x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y)/(1-x*y). G.f. of matrix power T^m: 1/(1-x*y)+ m*x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y)/(1-x*y).

EXAMPLE

Matrix inverse is: T^-1 = 2*I - T.

Matrix log is: log(T) = T - I.

Triangle T begins:

1, 1;

1, 0, 1;

1,-1, 1, 1;

1, 0, 0, 0, 1;

1,-1, 0, 0, 1, 1;

1, 0, 1, 0,-1, 0, 1;

1,-1,-1,-1, 1, 1, 1, 1;

1, 0, 2, 2, 0,-2,-2, 0, 1;

1,-1,-2,-4,-2, 2, 4, 2, 1, 1;

1, 0, 3, 6, 6, 0,-6,-6,-3, 0, 1;

1,-1,-3,-9,-12,-6, 6, 12, 9, 3, 1, 1;

1, 0, 4, 12, 21, 18, 0,-18,-21,-12,-4, 0, 1; ...

The g.f. of column k, C_k(x), obeys the recurrence:

C_k = C_{k-1} + (-1)^k*x*(1+2*x)/(1-x)/(1+x)^k with C_0 = 1/(1-x);

so that column g.f.s continue as:

C_1 = C_0 - x*(1+2*x)/(1-x)/(1+x),

C_2 = C_1 + x*(1+2*x)/(1-x)/(1+x)^2,

C_3 = C_2 - x*(1+2*x)/(1-x)/(1+x)^3, ...

PROG

(PARI) T(n, k)=local(x=X+X*O(X^n), y=Y+Y*O(Y^k)); polcoeff(polcoeff( 1/(1-x*y)+ x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y)/(1-x*y), n, X), k, Y)

(PARI) T(n, k)=local(M=matrix(n+1, n+1)); for(r=1, n+1, for(c=1, r, M[r, c]=if(r==c, 1, if(c==1, 1, if(c>1, (2*M^0-M)[r-1, c-1])+(2*M^0-M)[r-1, c])))); return(M[n+1, k+1])

CROSSREFS

Cf. A116466 (unsigned row sums), A116467 (row squared sums), A064310 (central terms); A112555 (variant).

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Feb 19 2006

STATUS

approved

A116467

Row squared sums of triangle A114700: a(n) = Sum_{k=0..n} A114700(n,k)^2.

+10
1

1, 2, 2, 4, 2, 4, 4, 8, 18, 52, 164, 544, 1852, 6416, 22520, 79888, 285938, 1031316, 3744628, 13676384, 50209860, 185190400, 685888736, 2549873880, 9511787452, 35592025904, 133559834064, 502494233128, 1895089009088, 7162963968712

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..29.

FORMULA

G.f.: A(x) = 2*(1-6*x+2*x^2+6*x^3+3*x^4)/(2-10*x-x^2)/(1-x)^2 + x*(2+x)*(1-4*x)/(2-10*x-x^2)/2*deriv(A(x)).

PROG

(PARI) {a(n)=sum(k=0, n, polcoeff(polcoeff(1/(1-x*y)+ x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y+x*O(x^n)+y*O(y^k))/(1-x*y), n, x), k, y)^2)}