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%I #25 Dec 09 2021 07:15:14
%S -1,0,1,2,3,4,20,216,2072,18880,177984,1805440,19935872,239445504,
%T 3113377280,43588830208,653836446720,10461393240064,177843710148608,
%U 3201186844016640,60822550184493056,1216451004043755520,25545471085755629568,562000363888584687616
%N Expansion of the e.g.f. (1 - 2*x - 2*log(1 - x) - exp(2*x)*(1 - x)^2) / 4 - 1.
%C For all p prime, a(p) == -1 (mod p).
%C For n > 1, a(n) == 0 (mod (n-1)).
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^(k-1)*ceiling(2^(k-2))*A106828(n, k).
%F a(n) ~ (n-1)!/2. - _Vaclav Kotesovec_, Dec 09 2021
%e E.g.f.: -1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 4*x^5/5! + 20*x^6/6! + 216*x^7/7! + 2072*x^8/8! + 18880*x^9/9! + ...
%e a(19) = Sum_{k=1..9} (-1)^(k-1)*ceiling(2^(k-2))*A106828(19, k) = 3201186844016640.
%e For k = 1, (-1)^(1-1)*ceiling(2^(1-2))*A106828(19, 1) == -1 (mod 19), because (-1)^(1-1)*ceiling(2^(1-2)) = 1 and A106828(19, 1) = (19-1)!
%e For k >= 2, (-1)^(k-1)*ceiling(2^(k-2))*A106828(19, k) == 0 (mod 19), because A106828(19, k) == 0 (mod 19), result a(19) == -1 (mod 19).
%e a(10) = Sum_{k=1..5} (-1)^(k-1)*ceiling(2^(k-2))*A106828(10, k) = 177984.
%e a(10) == 0 (mod (10-1)), because for k >= 1, A106828(10, k) == 0 (mod 9).
%p a := series((1-2*x-2*log(1-x)-exp(2*x)*(1-x)^2)/4-1, x=0, 25):
%p seq(n!*coeff(a, x, n), n=0..23);
%p # second program:
%p a := n -> add((-1)^(k-1)*ceil(2^(k-2))*A106828(n, k), k=0..iquo(n, 2)):
%p seq(a(n), n=0..23);
%t CoefficientList[Series[(1 - 2*x - 2*Log[1 - x] - E^(2*x)*(1 - x)^2)/4 - 1, {x, 0, 23}], x]*Range[0, 23]!
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace((1-2*x-2*log(1-x)-exp(2*x)*(1-x)^2)/4 - 1)) \\ _Michel Marcus_, Aug 23 2021
%Y Cf. A106828, A343482, A345697, A345969, A346119.
%K sign
%O 0,4
%A _Mélika Tebni_, Aug 23 2021
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