

A290540


Determinant of circulant matrix of order 10 with entries in the first row that are (1)^(j1)*Sum_{k>=0} (1)^k*binomial(n, 10*k+j1), for j=1..10.


1



1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2276485387658524, 523547340003805770400, 39617190432735671861429500, 2896792542975174202888623380000, 95819032881785191861991031568287500, 1018409199709889673458815786392849200000
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OFFSET

0,11


COMMENTS

a(n) = 0 for n == 9 (mod 10).
A generalization. For an even N >= 2, consider the determinant of circulant matrix of order N with entries in the first row (1)^(j1)K_j(n), j=1..N, where K_j(n) = Sum_{k>=0} (1)^k*binomial(n, N*k+j1). Then it is 0 for n == N1 (mod N). This statement follows from an easily proved identity K_j(N*t + N  1) = (1)^t*K_(N  j + 1)(N*t + N  1) and a known calculation formula for the determinant of circulant matrix [Wikipedia]. Besides, it is 0 for n=1..N2. We also conjecture that every such sequence contains infinitely many blocks of N1 negative and N1 positive terms separated by 0's.


LINKS

Table of n, a(n) for n=0..15.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Wikipedia, Circulant matrix


MAPLE

f:= n > LinearAlgebra:Determinant(Matrix(10, 10, shape=
Circulant[seq((1)^j*add((1)^k*binomial(n, 10*k+j),
k=0..(nj)/10), j=0..9)])):
map(f, [$0..20]); # Robert Israel, Aug 08 2017


MATHEMATICA

ro[n_] := Table[(1)^(j1) Sum[(1)^k Binomial[n, 10k+j1], {k, 0, n/10}], {j, 1, 10}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 9}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 15}] (* JeanFrançois Alcover, Aug 10 2018 *)


CROSSREFS

Cf. A290286, A290535, A290539.
Sequence in context: A047698 A246252 A058445 * A132910 A172550 A216908
Adjacent sequences: A290537 A290538 A290539 * A290541 A290542 A290543


KEYWORD

sign


AUTHOR

Vladimir Shevelev and Peter J. C. Moses, Aug 05 2017


STATUS

approved



