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%I #7 Jun 26 2015 21:38:22
%S 1,1,1,1,1,1,3,3,1,1,21,21,6,1,1,274,274,75,10,1,1,5806,5806,1565,195,
%T 15,1,1,182766,182766,48950,5940,420,21,1,1,8034916,8034916,2145626,
%U 257300,17570,798,28,1,1,471517614,471517614,125727238,14989472,1006880
%N Triangle T, read by rows, where column k of matrix power T^( k(k+1)/2 ) equals left-shifted column (k-1) of T for k>=1.
%C Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1. Also, column k equals unsigned column k of the matrix inverse of triangle P_k defined by P_k(m,j) = C( C(j+2,3) - C(k+2,3) + m-j, m-j) for m>=j>=0.
%F G.f. of column k: 1/(1-x) = Sum_{n>=0} T(n+k,k)*x^n*(1-x)^p_k(n), so that column k equals the number of subpartitions of the partition p_k defined by: p_k(n) = (n^2 + (3*k+3)*n + (3*k^2+6*k-4))*n/6 for n>=0.
%e Triangle T begins:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 3, 3, 1, 1;
%e 21, 21, 6, 1, 1;
%e 274, 274, 75, 10, 1, 1;
%e 5806, 5806, 1565, 195, 15, 1, 1;
%e 182766, 182766, 48950, 5940, 420, 21, 1, 1;
%e 8034916, 8034916, 2145626, 257300, 17570, 798, 28, 1, 1; ...
%e where column 1 of T^1 equals left-shifted column 0 of T.
%e Matrix cube T^3 begins:
%e 1;
%e 3, 1;
%e 6, 3, (1);
%e 22, 12, (3), 1;
%e 163, 91, (21), 3, 1;
%e 2167, 1219, (274), 33, 3, 1;
%e 46248, 26091, (5806), 661, 48, 3, 1;
%e 1460301, 824853, (182766), 20341, 1369, 66, 3, 1; ...
%e where column 2 of T^3 equals left-shifted column 1 of T.
%e Matrix power T^6 begins:
%e 1;
%e 6, 1;
%e 21, 6, 1;
%e 98, 33, 6, (1);
%e 791, 281, 51, (6), 1;
%e 10850, 3929, 710, (75), 6, 1;
%e 234472, 85557, 15425, (1565), 105, 6, 1;
%e 7444172, 2725402, 490806, (48950), 3080, 141, 6, 1; ...
%e where column 3 of T^6 equals left-shifted column 2 of T.
%o (PARI) {T(n,k)=abs((matrix(n+1,n+1,r,c, binomial((c-1)*c*(c+1)/3!-k*(k+1)*(k+2)/3!+r-c,r-c))^-1)[n+1,k+1])}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (PARI) /* As Defined by Matrix Product A126460 = A126445^-1*A126450: */
%o {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!,r-c))), N=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!+1,r-c)))); (M^-1*N)[n+1,k+1]}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y Columns: A126461, A126462, A126463, A126464; A126465 (dual); A107876 (variant); subpartitions defined: A115728.
%K nonn,tabl
%O 0,7
%A _Paul D. Hanna_, Dec 27 2006
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