A107865
Matrix inverse of triangle A107862.
Original entry on oeis.org
1, -1, 1, -1, -2, 1, -7, -4, -3, 1, -77, -26, -9, -4, 1, -1145, -287, -67, -16, -5, 1, -21410, -4412, -798, -139, -25, -6, 1, -481683, -86004, -13029, -1830, -251, -36, -7, 1, -12655196, -2017658, -268368, -32191, -3667, -412, -49, -8, 1, -379998938, -55134458, -6630228, -705680, -69868, -6657, -631
Offset: 0
Triangle begins:
1;
-1,1;
-1,-2,1;
-7,-4,-3,1;
-77,-26,-9,-4,1;
-1145,-287,-67,-16,-5,1;
-21410,-4412,-798,-139,-25,-6,1;
-481683,-86004,-13029,-1830,-251,-36,-7,1; ...
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{T(n,k)=(matrix(n+1,n+1,r,c,if(r>=c, binomial((r-1)*(r-2)/2-(c-1)*(c-2)/2+r-c,r-c)))^-1)[n+1,k+1]}
A107889
Triangular matrix T, read by rows, that satisfies: [T^-k](n,k) = -T(n,k-1) for n >= k > 0, or, equivalently, (column k of T^-k) = -SHIFT_LEFT(column k-1 of T) when zeros above the diagonal are ignored. Also, matrix inverse of triangle A107876.
Original entry on oeis.org
1, -1, 1, 0, -1, 1, 0, -1, -1, 1, 0, -3, -2, -1, 1, 0, -15, -9, -3, -1, 1, 0, -106, -61, -18, -4, -1, 1, 0, -975, -550, -154, -30, -5, -1, 1, 0, -11100, -6195, -1689, -310, -45, -6, -1, 1, 0, -151148, -83837, -22518, -4005, -545, -63, -7, -1, 1, 0, -2401365, -1326923, -353211, -61686, -8105, -875, -84, -8, -1, 1
Offset: 0
G.f. for column 1:
1 = T(1,1)*(1-x)^-1 + T(2,1)*x*(1-x)^0 + T(3,1)*x^2*(1-x)^2 + T(4,1)*x^3*(1-x)^5 + T(5,1)*x^4*(1-x)^9 + T(6,1)*x^5*(1-x)^14 + ...
= 1*(1-x)^-1 - 1*x*(1-x)^0 - 1*x^2*(1-x)^2 - 3*x^3*(1-x)^5 - 15*x^4*(1-x)^9 - 106*x^5*(1-x)^14 - 975*x^6*(1-x)^20 + ...
G.f. for column 2:
1 = T(2,2)*(1-x)^-1 + T(3,2)*x*(1-x)^1 + T(4,2)*x^2*(1-x)^4 + T(5,2)*x^3*(1-x)^8 + T(6,2)*x^4*(1-x)^13 + T(7,2)*x^5*(1-x)^19 + ...
= 1*(1-x)^-1 - 1*x*(1-x)^1 - 2*x^2*(1-x)^4 - 9*x^3*(1-x)^8 - 61*x^4*(1-x)^13 - 550*x^5*(1-x)^19 - 6195*x^6*(1-x)^26 + ...
Triangle begins:
1;
-1, 1;
0, -1, 1;
0, -1, -1, 1;
0, -3, -2, -1, 1;
0, -15, -9, -3, -1, 1;
0, -106, -61, -18, -4, -1, 1;
0, -975, -550, -154, -30, -5, -1, 1;
0, -11100, -6195, -1689, -310, -45, -6, -1, 1;
...
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max = 10;
A107862 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n - k], n - k], {n, 0, max}, {k, 0, max}];
A107867 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n-k+1], n - k], {n, 0, max}, {k, 0, max}];
T = Inverse[Inverse[A107862].A107867];
Table[T[[n + 1, k + 1]], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 31 2024 *)
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{T(n,k)=polcoeff(1-sum(j=0,n-k-1, T(j+k,k)*x^j*(1-x+x*O(x^n))^(-1+(k+j)*(k+j-1)/2-k*(k-1)/2)),n-k)}
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