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%I #12 Dec 09 2021 07:21:35
%S -1,0,0,0,-3,-20,-70,-84,1267,18824,209484,2284920,26010369,314864628,
%T 4073158102,56304102596,830061867975,13016975343184,216535182535928,
%U 3810394068301296,70744547160678501,1382375535029293500,28364229790262962386,609820072529413714012
%N a(n) = Sum_{k=0..floor(n/2)} (-1)^(k-1)*(k-1)^2*A106828(n, k).
%C For all p prime, a(p) == 0 (mod p*(p-1)).
%F E.g.f.: (-1 + 2*x - 2*x^2 + x^3 + (1 - x)*(log((1 - x)^(1 - 2*x)) - (log(1 - x))^2))*exp(x).
%F a(n) ~ 2 * exp(1) * log(n) * n! / n^2 * (1 + (gamma - 3/2)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Dec 09 2021
%e E.g.f.: -1 - 3*x^4/4! - 20*x^5/5! - 70*x^6/6! - 84*x^7/7! + 1267*x^8/8! + 18824*x^9/9! + ...
%e a(11) = Sum_{k=0..5} (-1)^(k-1)*(k-1)^2*A106828(11, k).
%e a(11) = (-1)*1*0 + (1)*0*3628800 + (-1)*1*6636960 + (1)*4*3678840 + (-1)*9*705320 + (1)*16*34650 = 2284920.
%e For k = 0, A106828(11,0) = 0.
%e For k = 1, (1-1)^2 = 0.
%e For 2 <= k <= 5, A106828(11, k) == 0 (mod 11*10).
%e Result a(11) == 0 (mod 11*10).
%p a := series((-1+2*x-2*x^2+x^3+(1-x)*(log((1-x)^(1-2*x))-(log(1-x))^2))*exp(x), x=0, 24):
%p seq(n!*coeff(a, x, n), n=0..23);
%p # second program:
%p a := n -> add((-1)^(k-1)*(k-1)^2*A106828(n, k), k=0..iquo(n, 2)):
%p seq(a(n), n=0..23);
%t CoefficientList[Series[(-1+2*x-2*x^2+x^3+(1-x)*(Log[(1-x)^(1-2*x)]-(Log[1-x])^2))*Exp[x], {x, 0, 23}], x]*Range[0, 23]!
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace((-1 + 2*x - 2*x^2 + x^3 + (1 - x)*(log((1 - x)^(1 - 2*x)) - (log(1 - x))^2))*exp(x))) \\ _Michel Marcus_, Oct 07 2021
%Y Cf. A106828, A347210, A347571.
%K sign
%O 0,5
%A _Mélika Tebni_, Oct 07 2021
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