%I #10 May 31 2024 10:40:43
%S 1,2,1,3,2,1,7,5,2,1,26,19,7,2,1,141,104,37,9,2,1,1034,766,268,61,11,
%T 2,1,9693,7197,2496,550,91,13,2,1,111522,82910,28612,6195,982,127,15,
%U 2,1,1528112,1136923,391189,83837,12977,1596,169,17,2,1,24372513,18141867,6230646,1326923,202494,24206,2424,217,19,2,1
%N Matrix square of triangle A107876; equals matrix product of triangles: A107876^2 = A107862^-1*A107870 = A107867^-1*A107873.
%C Column 0 is A107881. Column 1 is A107882. Column 3 equals A107883. Column 2 equals SHIFT_LEFT(A107877), where A107877 is column 1 of A107876.
%F G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(2+(k+j)*(k+j-1)/2-k*(k-1)/2).
%e G.f. for column 0:
%e 1 = T(0,0)*(1-x)^2 + T(1,0)*x*(1-x)^2 + T(2,0)*x^2*(1-x)^3 + T(3,0)*x^3*(1-x)^5 + T(4,0)*x^4*(1-x)^8 + T(5,0)*x^5*(1-x)^12 +...
%e = 1*(1-x)^2 + 2*x*(1-x)^2 + 3*x^2*(1-x)^3 + 7*x^3*(1-x)^5 + 26*x^4*(1-x)^8 + 141*x^5*(1-x)^12 +...
%e G.f. for column 1:
%e 1 = T(1,1)*(1-x)^2 + T(2,1)*x*(1-x)^3 + T(3,1)*x^2*(1-x)^5 + T(4,1)*x^3*(1-x)^8 + T(5,1)*x^4*(1-x)^12 + T(6,1)*x^5*(1-x)^17 +...
%e = 1*(1-x)^2 + 2*x*(1-x)^3 + 5*x^2*(1-x)^5 + 19*x^3*(1-x)^8 + 104*x^4*(1-x)^12 + 766*x^5*(1-x)^17 +...
%e Triangle T begins:
%e 1;
%e 2,1;
%e 3,2,1;
%e 7,5,2,1;
%e 26,19,7,2,1;
%e 141,104,37,9,2,1;
%e 1034,766,268,61,11,2,1;
%e 9693,7197,2496,550,91,13,2,1;
%e 111522,82910,28612,6195,982,127,15,2,1;
%e ...
%t max = 10;
%t 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}];
%t 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 T = MatrixPower[Inverse[A107862].A107867, 2];
%t Table[T[[n+1, k+1]], {n, 0, max}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 31 2024 *)
%o (PARI) {T(n,k)=polcoeff(1-sum(j=0,n-k-1, T(j+k,k)*x^j*(1-x+x*O(x^n))^(2+(k+j)*(k+j-1)/2-k*(k-1)/2)),n-k)}
%Y Cf. A107862, A107870, A107873, A107867, A107876, A107884, A107887.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Jun 04 2005