OFFSET
0,2
COMMENTS
FORMULA
G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(2+(k+j)*(k+j-1)/2-k*(k-1)/2).
EXAMPLE
G.f. for column 0:
1 = T(0,0)*(1-x)^2 + T(1,0)*x*(1-x)^2 + T(2,0)*x^2*(1-x)^3 + T(3,0)*x^3*(1-x)^5 + T(4,0)*x^4*(1-x)^8 + T(5,0)*x^5*(1-x)^12 +...
= 1*(1-x)^2 + 2*x*(1-x)^2 + 3*x^2*(1-x)^3 + 7*x^3*(1-x)^5 + 26*x^4*(1-x)^8 + 141*x^5*(1-x)^12 +...
G.f. for column 1:
1 = T(1,1)*(1-x)^2 + T(2,1)*x*(1-x)^3 + T(3,1)*x^2*(1-x)^5 + T(4,1)*x^3*(1-x)^8 + T(5,1)*x^4*(1-x)^12 + T(6,1)*x^5*(1-x)^17 +...
= 1*(1-x)^2 + 2*x*(1-x)^3 + 5*x^2*(1-x)^5 + 19*x^3*(1-x)^8 + 104*x^4*(1-x)^12 + 766*x^5*(1-x)^17 +...
Triangle T begins:
1;
2,1;
3,2,1;
7,5,2,1;
26,19,7,2,1;
141,104,37,9,2,1;
1034,766,268,61,11,2,1;
9693,7197,2496,550,91,13,2,1;
111522,82910,28612,6195,982,127,15,2,1;
...
MATHEMATICA
max = 10;
A107862 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n - k], n - k], {n, 0, max}, {k, 0, max}];
A107867 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n-k+1], n - k], {n, 0, max}, {k, 0, max}];
Table[T[[n+1, k+1]], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 31 2024 *)
PROG
(PARI) {T(n, k)=polcoeff(1-sum(j=0, n-k-1, T(j+k, k)*x^j*(1-x+x*O(x^n))^(2+(k+j)*(k+j-1)/2-k*(k-1)/2)), n-k)}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 04 2005
STATUS
approved