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%I #9 Dec 19 2021 11:07:20
%S 1,0,0,0,3,20,190,1764,17773,192632,2250036,28254600,380304639,
%T 5468906508,83750505826,1361579283596,23431400945145,425669127018416,
%U 8142731710207432,163636478165355408,3447201944202849819,75973975479088955460,1748531872985454054246,41951755708613404583732
%N a(n) = Sum_{k=0..floor(n/2)} (k-1)^2*A106828(n, k).
%C For all p prime, a(p) == 0 (mod p*(p-1)).
%F E.g.f.: (-2 - x + (3 + log((1 - x)^(1 + 2*x)) + (log(1 - x))^2) / (1 - x)) / exp(x).
%F a(n) ~ n! * exp(-1) * log(n)^2 * (1 + (2*gamma - 3)/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! + 190*x^6/6! + 1764*x^7/7! + 17773*x^8/8! + 192632*x^9/9! + ...
%e a(13) = Sum_{k=0..6} (k-1)^2*A106828(13, k).
%e a(13) = 1*0 + 0*479001600 + 1*967524480 + 4*647536032 + 9*177331440 + 16*18858840 + 25*540540 = 5468906508.
%e For k = 0, A106828(13, 0) = 0.
%e For k = 1, (1-1)^2 = 0.
%e For 2 <= k <= 6, A106828(13, k) == 0 (mod 13*12).
%e Result a(13) == 0 (mod 13*12).
%p a := n -> add((k-1)^2*A106828(n, k), k=0..iquo(n, 2)):
%p seq(a(n), n=0..23);
%p # second program:
%p a := series((-2-x+(3+log((1-x)^(1+2*x))+(log(1-x))^2)/(1-x))/exp(x), x=0, 24):
%p seq(n!*coeff(a, x, n), n=0..23);
%t CoefficientList[Series[(-2-x+(3+Log[(1-x)^(1+2*x)]+(Log[1-x])^2)/(1-x))/Exp[x], {x, 0, 23}], x]*Range[0, 23]!
%o (PARI) E2(n, m) = sum(k=0, n-m, (-1)^(n+k)*binomial(2*n+1, k)*stirling(2*n-m-k+1, n-m-k+1, 1)); \\ A008517
%o ast1(n, k) = if ((n==0) && (k==0), 1, sum(j=0, n-k, binomial(j, n-2*k)*E2(n-k, j+1))); \\ A106828
%o a(n) = sum(k=0, n\2, (k-1)^2*ast1(n, k)); \\ _Michel Marcus_, Dec 07 2021
%Y Cf. A106828, A347210, A347571, A348208.
%K nonn
%O 0,5
%A _Mélika Tebni_, Dec 07 2021
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