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%I #10 Jan 09 2024 12:17:29
%S 1,3,1,21,6,1,219,57,9,1,2973,723,111,12,1,49323,11361,1713,183,15,1,
%T 964173,212151,31575,3351,273,18,1,21680571,4584081,675489,71391,5799,
%U 381,21,1,551173053,112480887,16442823,1732881,140529,9219,507,24,1
%N Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((3*p-1)/3)) = (3*p-1)*(column p of T), or [T^((3*p-1)/3)](m,0) = (3*p-1)*T(p+m,p) for all m>=1 and p>=0.
%C Column 0 is A107716 (INVERTi of triple factorials). Column 1 is A107718 (twcie column 0 of T^(2/3), offset 1). The matrix logarithm divided by 3 yields the integer triangle A107724.
%F T(n, k) = 3*(k+1)*T(n, k+1) + Sum_{j=1..n-k-1} T(j, 0)*T(n, j+k+1) for n>k>=0, with T(n, n) = 1 for n>=0. T(n, 0) = A107716(n+1) for n>=0.
%e SHIFT_LEFT(column 0 of T^(p-1/3)) = (3*p-1)*(column p of T):
%e SHIFT_LEFT(column 0 of T^(-1/3)) = -1*(column 0 of T);
%e SHIFT_LEFT(column 0 of T^(2/3)) = 2*(column 1 of T);
%e SHIFT_LEFT(column 0 of T^(5/3)) = 5*(column 2 of T).
%e Triangle begins:
%e 1;
%e 3,1;
%e 21,6,1;
%e 219,57,9,1;
%e 2973,723,111,12,1;
%e 49323,11361,1713,183,15,1;
%e 964173,212151,31575,3351,273,18,1;
%e 21680571,4584081,675489,71391,5799,381,21,1; ...
%e Matrix power (2/3), T^(2/3), is A107719 and begins:
%e 1;
%e 2,1;
%e 12,4,1;
%e 114,32,6,1;
%e 1446,364,62,8,1;
%e 22722,5276,854,102,10,1; ...
%e compare column 0 of T^(2/3) to 2*(column 1 of T).
%e Matrix inverse cube-root T^(-1/3) is A107727 and begins:
%e 1;
%e -1,1;
%e -3,-2,1;
%e -21,-7,-3,1;
%e -219,-53,-13,-4,1;
%e -2973,-583,-115,-21,-5,1; ...
%e compare column 0 of T^(-1/3) to column 0 of T.
%e Matrix inverse is A107726 and begins:
%e 1;
%e -3,1;
%e -3,-6,1;
%e -21,-3,-9,1;
%e -219,-21,-3,-12,1;
%e -2973,-219,-21,-3,-15,1; ...
%e compare column 0 of T^(-1) to column 0 of T.
%o (PARI) {T(n,k)=if(n<k||k<0,0,if(n==k,1, 3*(k+1)*T(n,k+1)+sum(j=1,n-k-1,T(j,0)*T(n,j+k+1))))}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", ")); print(""))
%o (PARI) {T(n,k)=if(n<k||k<0,0,(matrix(n+1,n+1,m,j, if(m>=j,if(m==j,1,if(m==j+1,-3*j,-T(m-j-1,0)))))^-1)[n+1,k+1])}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", ")); print(""))
%Y Cf. A105615, A107716, A107718-A107727.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, May 30 2005
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