A102220 Triangular matrix, read by rows, equal to [2*I - A008459]^(-1), i.e., the matrix inverse of the difference of twice the identity matrix and the triangular matrix of squared binomial coefficients.

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, 87054773, 71134427, 14531391, 1319325, 67375, 2205, 49, 1, 6818444405, 5571505472, 1138150832, 103334336, 5277300, 172480, 3920, 64, 1

Offset: 0

Author

Paul D. Hanna, Dec 31 2004

Keywords

• nonn
• tabl

Comments

Column 0 forms A102221. Row sums form twice column 0 for n>0. Matrix logarithm is A102222.

Examples

			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],...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A008459, A102221, A102222.

Programs

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 *)

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]}

Formula

T(n,k) = C(n,k)^2*A102221(n-k). T(n,0) = A102221(n). 2*A102221(n) = Sum_{k=0..n} T(n,k) for n>0.