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Triangular matrix T, read by rows, such that the anticommutator of T and U shifts the columns of T up 1 row: {T,U}(n,k) = T(n+1,k), where U denotes the triangular matrix defined by U(n,k) = A000108(n-k) = Catalan(n-k) for n>=k and where T(n,n) = (n+1).
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%I #10 Jul 24 2019 02:03:33

%S 1,2,2,7,4,3,28,13,6,4,117,48,19,8,5,496,187,68,25,10,6,2110,748,257,

%T 88,31,12,7,8968,3034,1000,327,108,37,14,8,38017,12400,3958,1252,397,

%U 128,43,16,9,160648,50887,15832,4882,1504,467,148,49,18,10

%N Triangular matrix T, read by rows, such that the anticommutator of T and U shifts the columns of T up 1 row: {T,U}(n,k) = T(n+1,k), where U denotes the triangular matrix defined by U(n,k) = A000108(n-k) = Catalan(n-k) for n>=k and where T(n,n) = (n+1).

%C Difference between any two columns = A000984 (central binomial coefficients):

%C T(n+1,k+1) = T(n,k) + A000984(n-k).

%F G.f. of column k: k/sqrt(1-4*x) + 4*x^2/(1+sqrt(1-4*x))^2/(1-4*x)^(3/2).

%F G.f.: A(x,y) = 1/sqrt(1-4*x)/(1-x*y)^2 + 4*x^2/(1+sqrt(1-4*x))^2/(1-4*x)^(3/2)/(1-x*y).

%e Triangular matrix T begins:

%e 1;

%e 2,2;

%e 7,4,3;

%e 28,13,6,4;

%e 117,48,19,8,5;

%e 496,187,68,25,10,6;

%e 2110,748,257,88,31,12,7;

%e 8968,3034,1000,327,108,37,14,8;

%e 38017,12400,3958,1252,397,128,43,16,9; ...

%e Define triangular matrix U by U(n,k) = Catalan(n-k):

%e 1;

%e 1,1;

%e 2,1,1;

%e 5,2,1,1;

%e 14,5,2,1,1; ...

%e Then anticommutator {T,U} = T*U + U*T is given by

%e {T,U}(n,k) = T(n+1,k):

%e 2;

%e 7,4;

%e 28,13,6;

%e 117,48,19,8;

%e 496,187,68,25,10; ...

%e which equals T with columns shift up 1 row.

%e The commutator [T,U] = T*U - U*T is given by

%e [T,U](n,k) = 4^(n-k) - C(2*(n-k)+1,n-k):

%e 0,

%e 1,0,

%e 6,1,0,

%e 29,6,1,0,

%e 130,29,6,1,0, ...

%e Further, commutator [U^-1,T] = (U^-1)*T - T*(U^-1) is

%e [U^-1,T](n,k) = 4^(n-k-1) if n>k, else 0:

%e 0;

%e 1,0;

%e 4,1,0;

%e 16,4,1,0;

%e 64,16,4,1,0; ...

%t T[n_, k_] := SeriesCoefficient[1/Sqrt[1 - 4 x]/(1 - x y)^2 + 4 x^2/(1 + Sqrt[1 - 4 x])^2/(1 - 4 x)^(3/2)/(1 - x y), {x, 0, n}, {y, 0, k}];

%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 24 2019 *)

%o (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));polcoeff(polcoeff( 1/(1-X*Y)^2/sqrt(1-4*X)+4*X^2/(1+sqrt(1-4*X))^2/(1-4*X)^(3/2)/(1-X*Y),n,x),k,y)}

%Y Cf. A116078 (column 0), A000108, A000984.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Feb 04 2006