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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

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: A234462 A153853 A067000 * A168593 A328182 A157089

Adjacent sequences:  A307647 A307648 A307649 * A307651 A307652 A307653

KEYWORD

nonn

AUTHOR

Michel Marcus, Apr 20 2019

STATUS

approved

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Last modified December 6 22:42 EST 2021. Contains 349567 sequences. (Running on oeis4.)