A121435 - OEIS

%I #7 Jun 12 2018 22:44:23

%S 1,-1,1,1,-2,1,-2,5,-4,1,7,-19,18,-7,1,-37,104,-106,49,-11,1,268,-766,

%T 809,-406,110,-16,1,-2496,7197,-7746,4060,-1210,216,-22,1,28612,

%U -82910,90199,-48461,15235,-3032,385,-29,1,-391189,1136923,-1244891,678874,-220352,46732,-6699,638,-37,1

%N Matrix inverse of triangle A122175, where A122175(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.

%F (1) T(n,k) = A121434(n-1,k) - A121434(n-1,k+1).

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

%F G.f.s for column k:

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

%F (4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 1).

%e Triangle begins:

%e 1;

%e -1, 1;

%e 1, -2, 1;

%e -2, 5, -4, 1;

%e 7, -19, 18, -7, 1;

%e -37, 104, -106, 49, -11, 1;

%e 268, -766, 809, -406, 110, -16, 1;

%e -2496, 7197, -7746, 4060, -1210, 216, -22, 1;

%e 28612, -82910, 90199, -48461, 15235, -3032, 385, -29, 1;

%e -391189, 1136923, -1244891, 678874, -220352, 46732, -6699, 638, -37, 1; ...

%o (PARI) /* Matrix Inverse of A122175 */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r-1,r-c)))); return((M^-1)[n+1,k+1])

%o (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+1)), n-k)

%Y Cf. A098568, A107876; unsigned columns: A107877, A107882.

%K sign,tabl

%O 0,5

%A _Paul D. Hanna_, Aug 27 2006