A121434 Matrix inverse of triangle A098568, where A098568(n, k) = C( (k+1)*(k+2)/2 + n-k-1, n-k) for n>=k>=0.

1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -7, 12, -6, 1, 0, 37, -67, 39, -10, 1, 0, -268, 498, -311, 95, -15, 1, 0, 2496, -4701, 3045, -1015, 195, -21, 1, 0, -28612, 54298, -35901, 12560, -2675, 357, -28, 1, 0, 391189, -745734, 499157, -179717, 40635, -6097, 602, -36, 1, 0, -6230646, 11911221, -8034267, 2945010

Offset: 0

Author

Paul D. Hanna, Aug 27 2006

Keywords

• sign
• tabl

Examples

			Triangle begins:
  1;
  0, 1;
  0, -1, 1;
  0, 2, -3, 1;
  0, -7, 12, -6, 1;
  0, 37, -67, 39, -10, 1;
  0, -268, 498, -311, 95, -15, 1;
  0, 2496, -4701, 3045, -1015, 195, -21, 1;
  0, -28612, 54298, -35901, 12560, -2675, 357, -28, 1;
  0, 391189, -745734, 499157, -179717, 40635, -6097, 602, -36, 1; ...
		

Crossrefs

Cf. A098568, A107876; unsigned columns: A107877, A107887.

Programs

PARI

/* Matrix Inverse of A098568 */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r-2,r-c)))); return((M^-1)[n+1,k+1])

PARI

/* Obtain by G.F. */ T(n,k)=polcoeff(1-sum(j=0, n-k-1, T(j+k,k)*x^j/(1-x+x*O(x^n))^(j*(j+1)/2+j*k+k*(k+1)/2)), n-k)

Formula

(1) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2)](n,k); i.e., column k equals signed column k of A107876^(k*(k+1)/2).

G.f.s for column k:

(2) 1 = Sum_{j>=0} T(j+k,k)*x^j/(1-x)^( j*(j+1)/2) + j*k + k*(k+1)/2);

(3) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2).