A107717 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.
1, 3, 1, 21, 6, 1, 219, 57, 9, 1, 2973, 723, 111, 12, 1, 49323, 11361, 1713, 183, 15, 1, 964173, 212151, 31575, 3351, 273, 18, 1, 21680571, 4584081, 675489, 71391, 5799, 381, 21, 1, 551173053, 112480887, 16442823, 1732881, 140529, 9219, 507, 24, 1
Offset: 0
Examples
SHIFT_LEFT(column 0 of T^(p-1/3)) = (3*p-1)*(column p of T): SHIFT_LEFT(column 0 of T^(-1/3)) = -1*(column 0 of T); SHIFT_LEFT(column 0 of T^(2/3)) = 2*(column 1 of T); SHIFT_LEFT(column 0 of T^(5/3)) = 5*(column 2 of T). Triangle begins: 1; 3,1; 21,6,1; 219,57,9,1; 2973,723,111,12,1; 49323,11361,1713,183,15,1; 964173,212151,31575,3351,273,18,1; 21680571,4584081,675489,71391,5799,381,21,1; ... Matrix power (2/3), T^(2/3), is A107719 and begins: 1; 2,1; 12,4,1; 114,32,6,1; 1446,364,62,8,1; 22722,5276,854,102,10,1; ... compare column 0 of T^(2/3) to 2*(column 1 of T). Matrix inverse cube-root T^(-1/3) is A107727 and begins: 1; -1,1; -3,-2,1; -21,-7,-3,1; -219,-53,-13,-4,1; -2973,-583,-115,-21,-5,1; ... compare column 0 of T^(-1/3) to column 0 of T. Matrix inverse is A107726 and begins: 1; -3,1; -3,-6,1; -21,-3,-9,1; -219,-21,-3,-12,1; -2973,-219,-21,-3,-15,1; ... compare column 0 of T^(-1) to column 0 of T.
Programs
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PARI
{T(n,k)=if(n -
PARI
{T(n,k)=if(n
Formula
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.
Comments