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

0,2

Comments

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.

Links

Table of n, a(n) for n=0..44.

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.

Example

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.

Prog

(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))))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(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])}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A105615, A107716, A107718-A107727.

Sequence in context: A221365 A144279 A144280 * A143173 A000369 A225471

Adjacent sequences: A107714 A107715 A107716 * A107718 A107719 A107720

Keyword

nonn,tabl

Author

Paul D. Hanna, May 30 2005

Status

approved