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A336292
a(n) = (n!)^2 * Sum_{k=1..n} (-1)^(n-k) / (k * ((n-k)!)^2).
2
0, 1, -2, 3, 8, 305, 10734, 502747, 30344992, 2307890097, 216571514030, 24619605092291, 3337294343698248, 532148381719443073, 98646472269855762238, 21041945289232131607995, 5118447176652195630775424, 1408601897794844346184122017, 435481794298015565250651718302
OFFSET
0,3
FORMULA
Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(1 - x) * BesselJ(0,2*sqrt(x)).
MATHEMATICA
Table[(n!)^2 Sum[(-1)^(n - k)/(k ((n - k)!)^2), {k, 1, n}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[1 - x] BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
PROG
(PARI) a(n) = (n!)^2 * sum(k=1, n, (-1)^(n-k) / (k * ((n-k)!)^2)); \\ Michel Marcus, Jul 17 2020
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Jul 16 2020
STATUS
approved