OFFSET
0,2
COMMENTS
FORMULA
T(n, k) = 2*(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) = A000698(n+1) for n>=0.
EXAMPLE
SHIFT_LEFT(column 0 of T^(-1/2)) = -1*(column 0 of T);
SHIFT_LEFT(column 0 of T^(1/2)) = 1*(column 1 of T);
SHIFT_LEFT(column 0 of T^(3/2)) = 3*(column 2 of T);
SHIFT_LEFT(column 0 of T^(5/2)) = 5*(column 3 of T).
Triangle begins:
1;
2,1;
10,4,1;
74,26,6,1;
706,226,50,8,1;
8162,2426,522,82,10,1;
110410,30826,6498,1010,122,12,1;
1708394,451586,93666,14458,1738,170,14,1;
29752066,7489426,1532970,235466,28226,2754,226,16,1; ...
Matrix square-root T^(1/2) is A105623 which begins:
1;
1,1;
4,2,1;
26,10,3,1;
226,74,19,4,1;
2426,706,167,31,5,1; ...
compare column 0 of T^(1/2) to column 1 of T;
also, column 1 of T^(1/2) equals column 0 of T.
Matrix inverse square-root T^(-1/2) is A105620 which begins:
1;
-1,1;
-2,-2,1;
-10,-4,-3,1;
-74,-20,-7,-4,1;
-706,-148,-39,-11,-5,1; ...
compare column 0 of T^(-1/2) to column 0 of T.
Matrix inverse T^-1 is A105619 which begins:
1;
-2,1;
-2,-4,1;
-10,-2,-6,1;
-74,-10,-2,-8,1;
-706,-74,-10,-2,-10,1;
-8162,-706,-74,-10,-2,-12,1; ...
PROG
(PARI) {T(n, k) = if(n<k||k<0, 0, if(n==k, 1, 2*(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, -2*j, polcoeff(1/sum(i=0, m-j, (2*i)!/i!/2^i*x^i)+O(x^m), m-j)))))^-1)[n+1, k+1])}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 16 2005
STATUS
approved