OFFSET
0,4
COMMENTS
LINKS
Seiichi Manyama, Rows n = 0..139, flattened
FORMULA
T(n,k) = the (k+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = i, i >= 1 and 1 <= j <= i+1, and M[i,j] = 0, i >= 1 and j >= i+2, see the examples.
It appears that T(n,k) = (2*n-k)!/(2^(n-k)*(n-k)!) with conjectural e.g.f. 1/(x*(1-2*z) + (1-x)*sqrt(1-2*z)) = 1 + (1+x)*z + (3+3*x+2*x^2)*z^2/2! + .... Cf. A102625. - Peter Bala, Jul 09 2012
EXAMPLE
The first few rows of matrix M[i,j] are:
1, 1, 0, 0, 0, 0, ...
2, 2, 2, 0, 0, 0, ...
3, 3, 3, 3, 0, 0, ...
4, 4, 4, 4, 4, 0, ...
5, 5, 5, 5, 5, 5, ...
The first few rows of triangle T(n,k) are:
1;
1, 1;
3, 3, 2;
15, 15, 12, 6;
105, 105, 90, 60, 24;
945, 945, 840, 630, 360, 120;
10395, 10395, 9450, 7560, 5040, 2520, 720;
135135, 135135, 124740, 103950, 75600, 45360, 20160, 5040;
MAPLE
nmax:=7: M := Matrix(1..nmax+1, 1..nmax+1): for i from 1 to nmax do for j from 1 to i+1 do M[i, j] := i od: od: for n from 0 to nmax do B := M^n: for k from 0 to n do T(n, k) := B[1, k+1] od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od: seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jul 21 2011
PROG
(PARI) row(n)=(matrix(n, n, i, j, (i>j-2)*i)^(n-1))[1, ] \\ M. F. Hasler, Jul 24 2011
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jul 18 2011
EXTENSIONS
Corrected, edited and extended by Johannes W. Meijer, Jul 21 2011
More terms from Seiichi Manyama, Apr 06 2019
STATUS
approved