OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..110 (terms 0..69 from Alois P. Heinz)
FORMULA
a(n) = (4n)! * [x^(4n) t^(2n)] t^2/(1-t*x-(1-t^2)*exp(-t*x)). [corrected by Vaclav Kotesovec, Jul 09 2019]
a(n) = A097591(4n,2n).
From Vaclav Kotesovec, Jul 09 2019: (Start)
a(n)/(4*n)! ~ c * d^n / sqrt(n), where
d = 0.49313160144517183347479521733129940030484540928084707469774969650583707...
c = 3.44699229707824751737600849250650265725079793249740793784564520854062204...
a(n) ~ c * d^n * n^(4*n), where
d = 2.31219720619339615667811172118287009649702081583503593066663730992576726...
c = 17.2806567085831933774093124549232969200598807738253988225436890867215712...
(End)
EXAMPLE
a(1) = 17: (124)(3), (134)(2), 14(3)(2), (2)(134), (2)14(3), (234)(1), 24(3)(1), (3)(124), (3)14(2), (3)(2)14, (3)24(1), 34(2)(1), (4)(123), (4)13(2), (4)(2)13, (4)23(1), (4)(3)12; (odd length runs are shown between parentheses).
MATHEMATICA
Flatten[{1, Table[(4 n)! * Coefficient[Expand[Normal[Series[t^2/(1 - t*x - (1 - t^2)*E^(-t*x)), {x, 0, 4*n}, {t, 0, 2*n}]]], x^(4*n)*t^(2*n)], {n, 1, 10}]}] (* Vaclav Kotesovec, Jul 09 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 03 2019
STATUS
approved