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a(n) = numerator of -(1/4)*n!*(2 + n!)*(-2 + 1/(1 + floor(n/2 - 1/2))) - n!*Sum_{m=1..1 + 2*floor(n/2 - 1/2)} 1/m.
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%I #20 Jan 08 2024 22:44:50

%S -1,0,7,190,5826,214956,11104542,711175536,59256152496,5925678248160,

%T 730285755406560,105161159860398720,18003044434808914560,

%U 3528596711774282883840,801568243461355261718400,205201470326854119387494400,59742508072063053997776844800

%N a(n) = numerator of -(1/4)*n!*(2 + n!)*(-2 + 1/(1 + floor(n/2 - 1/2))) - n!*Sum_{m=1..1 + 2*floor(n/2 - 1/2)} 1/m.

%C In the sum formula below, changing n! to n in the outer summation yields A161664.

%F For n>1: a(n) = Sum_{h=1..n!} Sum_{m=1..1 + 2*floor(n/2 - 1/2)} Sum_{k=1 + floor(h/(m + 1))..floor(h/m - 1/m)} 1.

%e The fractions, of which a(n) is the numerator, begin: -1/4, 0, 7, 190, 5826, ...

%t Numerator[Table[-1/4*n!*(2 + n!)*(-2 + 1/(1 + Floor[n/2 - 1/2])) - n!*Sum[1/m, {m, 1, 1 + 2*Floor[n/2 - 1/2]}], {n, 1, 17}]]

%Y Cf. A161664, A006218.

%K sign,frac

%O 1,3

%A _Mats Granvik_, Dec 31 2023