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%I #11 Dec 06 2019 11:51:00
%S 1,1,1,5,4,1,55,45,9,1,1077,880,180,16,1,32951,26925,5500,500,25,1,
%T 1451723,1186236,242325,22000,1125,36,1,87054773,71134427,14531391,
%U 1319325,67375,2205,49,1,6818444405,5571505472,1138150832,103334336,5277300,172480,3920,64,1
%N 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.
%C Column 0 forms A102221. Row sums form twice column 0 for n>0. Matrix logarithm is A102222.
%H Alois P. Heinz, <a href="/A102220/b102220.txt">Rows n = 0..140, flattened</a>
%F 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.
%e Rows begin:
%e [1],
%e [1,1],
%e [5,4,1],
%e [55,45,9,1],
%e [1077,880,180,16,1],
%e [32951,26925,5500,500,25,1],
%e [1451723,1186236,242325,22000,1125,36,1],...
%e and equal the term-by-term product of column 0
%e with the squared binomial coefficients (A008459):
%e [(1)1^2],
%e [(1)1^2,(1)1^2],
%e [(5)1^2,(1)2^2,(1)1^2],
%e [(55)1^2,(5)3^2,(1)3^2,(1)1^2],
%e [(1077)1^2,(55)4^2,(5)6^2,(1)4^2,(1)1^2],...
%e The matrix inverse is [2*I - A008459]:
%e [1],
%e [ -1,1],
%e [ -1,-4,1],
%e [ -1,-9,-9,1],
%e [ -1,-16,-36,-16,1],...
%p b:= proc(n) option remember; `if`(n=0, 1,
%p add(b(n-i)*binomial(n, i)/i!, i=1..n))
%p end:
%p T:= (n, k)-> binomial(n, k)^2*b(n-k)*(n-k)!:
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Sep 10 2019
%t nmax = 10;
%t M = Inverse[2 IdentityMatrix[nmax+1] - Table[Binomial[n, k]^2, {n, 0, nmax}, {k, 0, nmax}]];
%t T[n_, k_] := M[[n+1, k+1]];
%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 06 2019 *)
%o (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]}
%Y Cf. A008459, A102221, A102222.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Dec 31 2004
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