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A347571 Expansion of the e.g.f. (-1 - 2*x - 2*log(1 - x) + exp(-2*x) / (1 - x)^2) / 4 + 1.

%I A347571 #8 Sep 10 2021 14:22:13
%S A347571 1,0,1,2,9,44,280,2064,17528,167488,1777536,20721920,263055232,
%T A347571 3610443264,53256280064,839974309888,14103897738240,251146689069056,
%U A347571 4726795773018112,93746994502828032,1954053073794596864,42702893781890498560,976276451410488066048,23303485413254033309696
%N A347571 Expansion of the e.g.f. (-1 - 2*x - 2*log(1 - x) + exp(-2*x) / (1 - x)^2) / 4 + 1.
%C A347571 For all p prime, a(p) == -1 (mod p).
%C A347571 For n > 1, a(n) == 0 (mod (n-1)).
%F A347571 a(n) = Sum_{k=0..floor(n/2)} ceiling(2^(k-2))*A106828(n, k).
%F A347571 a(n) ~ n * n! / (4*exp(2)). - _Vaclav Kotesovec_, Sep 10 2021
%e A347571 E.g.f.: 1 + x^2/2! + 2*x^3/3! + 9*x^4/4! + 44*x^5/5! + 280*x^6/6! + 2064*x^7/7! + 17528*x^8/8! + 167488*x^9/9! + ...
%e A347571 a(11) = Sum_{k=0..5} ceiling(2^(k-2))*A106828(11, k) = 20721920.
%e A347571 For k = 0, A106828(11,0) = 0.
%e A347571 For k = 1, ceiling(2^(1-2))*A106828(11, 1) == -1 (mod 11), because ceiling(2^(1-2)) = 1 and A106828(11, 1) = (11-1)!
%e A347571 For k >= 2, ceiling(2^(k-2))*A106828(11, k) == 0 (mod 11), because A106828(11, k) == 0 (mod 11), result a(11) == -1 (mod 11).
%e A347571 a(10) = Sum_{k=0..5} ceiling(2^(k-2))*A106828(10, k) = 1777536.
%e A347571 a(10) == 0 (mod (10-1)), because for k >= 0, A106828(10, k) == 0 (mod 9).
%p A347571 a := series((-1-2*x-2*log(1-x)+exp(-2*x)/(1-x)^2)/4+1, x=0, 24):
%p A347571 seq(n!*coeff(a, x, n), n=0..23);
%p A347571 # second program:
%p A347571 a := n -> add(ceil(2^(k-2))*A106828(n, k), k=0..iquo(n, 2)):
%p A347571 seq(a(n), n=0..23);
%t A347571 CoefficientList[Series[(-1 - 2*x - 2*Log[1 - x] + Exp[-2*x]/(1 - x)^2)/4 + 1, {x, 0, 23}], x]*Range[0, 23]!
%o A347571 (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_, Sep 07 2021
%Y A347571 Cf. A106828, A343482, A345697, A345969, A346119, A347210.
%K A347571 nonn
%O A347571 0,4
%A A347571 _Mélika Tebni_, Sep 07 2021