OFFSET
0,4
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
EXAMPLE
Rows begin:
[1],
[1,1],
[5,4,1],
[55,45,9,1],
[1077,880,180,16,1],
[32951,26925,5500,500,25,1],
[1451723,1186236,242325,22000,1125,36,1],...
and equal the term-by-term product of column 0
with the squared binomial coefficients (A008459):
[(1)1^2],
[(1)1^2,(1)1^2],
[(5)1^2,(1)2^2,(1)1^2],
[(55)1^2,(5)3^2,(1)3^2,(1)1^2],
[(1077)1^2,(55)4^2,(5)6^2,(1)4^2,(1)1^2],...
The matrix inverse is [2*I - A008459]:
[1],
[ -1,1],
[ -1,-4,1],
[ -1,-9,-9,1],
[ -1,-16,-36,-16,1],...
MAPLE
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)*binomial(n, i)/i!, i=1..n))
end:
T:= (n, k)-> binomial(n, k)^2*b(n-k)*(n-k)!:
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Sep 10 2019
MATHEMATICA
nmax = 10;
M = Inverse[2 IdentityMatrix[nmax+1] - Table[Binomial[n, k]^2, {n, 0, nmax}, {k, 0, nmax}]];
T[n_, k_] := M[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)
PROG
(PARI) {T(n, k)=(matrix(n+1, n+1, i, j, if(i==j, 2, 0)-binomial(i-1, j-1)^2)^-1)[n+1, k+1]}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 31 2004
STATUS
approved