OFFSET
0,3
COMMENTS
FORMULA
G.f. for column k: T(k, k) = k+1 = Sum_{n>=0} T(n+k, k)*x^n*Product_{j=k..[n/2+k]} (1-(j+1)*x).
EXAMPLE
Rows begin:
[1],
[1,2],
[1,4,3],
[3,8,9,4],
[7,40,27,16,5],
[36,152,189,64,25,6],
[139,1128,999,576,125,36,7],
[1036,6200,9720,3904,1375,216,49,8],
[5711,61120,69687,47040,11375,2808,343,64,9],...
The antidiagonals are formed by interleaving the
rows of triangle A102098:
[1],
[1,2],
[7,8,3],
[139,152,27,4],...
with the rows of the matrix square of A102098,
which is triangle A102920:
[1],
[3,4],
[36,40,9],
[1036,1128,189,16],...
G.f. for Column 0 (A102917): 1 = 1*(1-x) + 1*x*(1-x)
+ 1*x^2*(1-x)(1-2x) + 3*x^3*(1-x)(1-2x)
+ 7*x^4*(1-x)(1-2x)(1-3x) + 36*x^5*(1-x)(1-2x)(1-3x) +...
+ A082162(n)*x^(2n)*(1-x)(1-2x)*..*(1-(n+1)x)
+ A102921(n)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+1)x) + ...
G.f. for Column 1 (A102918): 2 = 2*(1-2x) + 4*x*(1-2x)
+ 8*x^2*(1-2x)(1-3x) + 40*x^3*(1-2x)(1-3x)
+ 152*x^4*(1-2x)(1-3x)(1-4x) + 1128*x^5*(1-2x)(1-3x)(1-4x) +...
+ T(2n+1,1)*x^(2n)*(1-2x)(1-3x)*..*(1-(n+2)x)
+ T(2n+2,1)*x^(2n+1)*(1-2x)(1-3x)*..*(1-(n+2)x) + ...
PROG
(PARI) {T(n, k)=if(n<k, 0, if(n==k, k+1, polcoeff(k+1-sum(i=k, n-1, T(i, k)*x^i*prod(j=1, (i-k)\2+1, 1-(j+k)*x+x*O(x^n))), n)))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 21 2005
STATUS
approved