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A307650
Number of regions in the Shi-threshold arrangement STn.
0
1, 1, 3, 27, 345, 5513, 106619, 2426819, 63595653, 1886225589, 62471885847, 2285488346111, 91536241567505, 3983627374435361, 187189099298640435, 9445664642509417563, 509426043718487709501, 29243389264220375989229, 1780261345776857866401263, 114560896953217093009807895
OFFSET
0,3
LINKS
A. R. Balasubramanian, Generalized threshold arrangements, arXiv:1904.08903 [math.CO], 2019. Says that sequence is 1, 27, 345, 5513, ... for n>=2.
Seunghyun Seo, Shi threshold arrangement, Electronic Journal of Combinatorics, volume 19, issue 3, 2012.
FORMULA
a(n) = Sum_{j>=0} binomial(2*j+1, j)*j!*(-1)^(n-j)*Stirling2(n,j) + 2*n*Sum_{j>=0} binomial(2*j+2, j)*j!*(-1)^(n-j)*Stirling2(n-1,j) + n*(n-1)*Sum_{j>=0} binomial(2*j+3, j)*j!*(-1)^(n-j)*Stirling2(n-2,j).
E.g.f.: C(1-exp(-x))*(x*C(1-exp(-x))-1)^2/sqrt(4*exp(-x)-3) where C(z) = (1-sqrt(1-4*z))/(2*z) is the g.f. for Catalan numbers.
PROG
(PARI) a(n) = sum(j=0, n, binomial(2*j+1, j)*j!*(-1)^(n-j)*stirling(n, j, 2)) + 2*n*sum(j=0, n-1, binomial(2*j+2, j)*j!*(-1)^(n-j)*stirling(n-1, j, 2)) + n*(n-1)*sum(j=0, n-2, binomial(2*j+3, j)*j!*(-1)^(n-j)*stirling(n-2, j, 2));
CROSSREFS
Cf. A000108 (Catalan numbers), A008277 (Stirling2).
Sequence in context: A153853 A067000 A354658 * A168593 A364431 A328182
KEYWORD
nonn
AUTHOR
Michel Marcus, Apr 20 2019
STATUS
approved