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%I #10 Mar 19 2013 18:08:15
%S 1,-2,1,3,-3,1,-7,9,-5,1,26,-37,25,-8,1,-141,210,-155,60,-12,1,1034,
%T -1575,1215,-516,126,-17,1,-9693,14943,-11806,5270,-1426,238,-23,1,
%U 111522,-173109,138660,-63696,18267,-3417,414,-30,1,-1528112,2381814,-1923765,899226,-267084,53431,-7337,675,-38,1
%N Matrix inverse of triangle A122176, where A122176(n,k) = C( k*(k+1)/2 + n-k + 1, n-k) for n>=k>=0.
%H Paul D. Hanna, <a href="/A121436/b121436.txt">Rows n=0..45, as a table of n, a(n) for n=0..1080.</a>
%F (1) T(n,k) = A121435(n-1,k) - A121435(n-1,k+1).
%F (2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 2)](n,k);
%F i.e., column k equals signed column k of A107876^(k*(k+1)/2 + 2).
%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 + 2);
%F (4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 2).
%e Triangle begins:
%e 1;
%e -2, 1;
%e 3, -3, 1;
%e -7, 9, -5, 1;
%e 26, -37, 25, -8, 1;
%e -141, 210, -155, 60, -12, 1;
%e 1034, -1575, 1215, -516, 126, -17, 1;
%e -9693, 14943, -11806, 5270, -1426, 238, -23, 1;
%e 111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1;
%e -1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1; ...
%o (PARI) /* Matrix Inverse of A122176 */
%o {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r,r-c)))); return((M^-1)[n+1,k+1])}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (PARI) /* Obtain by G.F. */
%o {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+2)), n-k)}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y Cf. A098568, A107876; unsigned columns: A107881, A107886.
%K sign,tabl
%O 0,2
%A _Paul D. Hanna_, Aug 27 2006
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