A145570 Circulants of Catalan numbers.

1, 0, 4, 459, 474743, 5237087765, 686666209113536, 1140713637429903585344, 24957776794187383667855422048, 7377122100200717681983830999516060000, 30004208141654594144715773978429859682880072414, 1703184026083327296951313841743251806796128938200000000000, 1365027457901516492029047382022588117973123824294791438142988114734512

Offset: 1

Author

Wolfdieter Lang Feb 05 2009

Keywords

• nonn
• easy

Comments

A circulant C_n is the determinant of a circular n X n matrix M, i.e. one with entries M_{i,j}=a_{i-j} where the indices are taken mod n. Hence C_n=C_n([a_n=a_0,a_{n-1},...,a_1]), with the first row of M given. The second row is [a_1,a_0,a_{n-1},...,a_2], etc.

The eigenvalues of a circular n X n matrix M(n) are lambda^{(n)}_k=sum(a_j*(rho_n)^(j*k),j=1..n), with the n-th roots of unity (rho_n)^k, k=1..n, where rho_n:=exp(2*Pi/n). See the P. J. Davis reference which uses a different convention, and the term circulant for circular.

Examples

			n=4: the circular 4 X 4 matrix is M(4) = matrix([[5,2,1,1],[1,5,2,1],[1,1,5,2],[2,1,1,5]]).
n=4: 4th roots of unity: rho_4 = I, (rho_4)^2 = -1, (rho_4)^3 = -I, (rho_4)^4 =1, with I^2=-1.
n=4: the eigenvalues of M(4) are therefore: 1*I^k + 1*(-1)^k + 2*(-I)^k + 5*1^k, k=1,..,4, namely 4-I, 3, 4+I, 9.
n=4: a(4)= Det(M(4)) = (4-I)*3*(4+I)*9 = 459.
		

References

• P. J. Davis, Circulant Matrices, J. Wiley, New York, 1979.

Crossrefs

A123744, A123745 (circulants for Fibonacci numbers).

Programs

Mathematica

rho[n_] := Exp[2*I*Pi/n]; lambda[n_, k_] := Sum[ CatalanNumber[j - 1]*rho[n]^(j*k), {j, 1, n}]; a[n_?EvenQ] := FullSimplify[ Product[ lambda[n, k], {k, 1, n}]]; a[n_?OddQ] := Expand[ Product[ lambda[n, k], {k, 1, n}]] /. Plus[x_Integer, Times[y_Integer, Power[E, Times[ Complex[0, Rational[, FactorInteger[n][[1, 1]]]], Pi]]], _] -> x - y; Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Sep 27 2011 *)

Formula

a(n)=product(lambda^{(n)}k,k=1..n), with lambda^{(n)}_k=sum(Ca{j-1}*(rho_n)^(j*k), j=1..n).

a(n) = C_n([Ca_{n-1},Ca_{n-2},...,Ca_0]) with the Catalan numbers Ca_n:=A000108(n), and the circulant C_n (see comment above).