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A226157 a(n) = BS2(n) * W(n) where BS2 = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) and S_{2}(n, k) are the Stirling-Frobenius subset numbers A039755(n, k). W(n) = product{p primes <= n+1 such that p divides n+1 or p-1 divides n} = A225481(n).

%I A226157 #20 Jun 27 2019 08:40:20
%S A226157 1,1,-2,-2,14,33,-62,-132,254,14585,-5110,-313266,2828954,38669001,
%T A226157 -573370,-404801672,237036478,117650567067,-11499383114,
%U A226157 -24255028327410,1281647882998,8203584532193105,-3584085584926,-418397193140056356,3965530936622474,405233976502715850633
%N A226157 a(n) = BS2(n) * W(n) where BS2 = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) and S_{2}(n, k) are the Stirling-Frobenius subset numbers A039755(n, k). W(n) = product{p primes <= n+1 such that p divides n+1 or p-1 divides n} = A225481(n).
%C A226157 a(n)/A225481(n) is case m = 2 of the scaled generalized Bernoulli numbers defined as Sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{m}(n, k) where S_{m}(n, k) are Stirling-Frobenius subset numbers. A225481(n) can be seen as an analog of the Clausen numbers A141056(n).
%H A226157 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">Stirling-Frobenius numbers</a>
%H A226157 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedBernoulliNumbers.html">Generalized Bernoulli numbers</a>.
%e A226157 The numerators of 1/1, 1/2, -2/6, -2/2, 14/30, 33/6, -62/42, -132/2, 254/30, 14585/10, -5110/66, ...(the denominators are A225481(n)).
%t A226157 EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = If[n == 0, If[k == 0, 1 , 0], (m*(n-k) + m - 1)*EulerianNumber[n-1, k-1, m] + (m*k + 1)* EulerianNumber[n-1, k, m]];
%t A226157 BS[n_, m_] := Sum[Sum[EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/ ((-m)^k*(k+1)), {k, 0, n}]
%t A226157 a[n_] := Product[If[Divisible[n+1, p] || Divisible[n, p-1], p, 1], {p, Prime /@ Range @ PrimePi[n+1]}] * BS[n, 2];
%t A226157 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 27 2019, from Sage *)
%o A226157 (Sage)
%o A226157 @CachedFunction
%o A226157 def EulerianNumber(n, k, m) :   # The Eulerian numbers
%o A226157     if n == 0: return 1 if k == 0 else 0
%o A226157     return ((m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m) +
%o A226157            (m*k+1)*EulerianNumber(n-1, k, m))
%o A226157 @CachedFunction
%o A226157 def BS(n, m):   # The generalized scaled Bernoulli numbers
%o A226157     return (add(add(EulerianNumber(n, j, m)*binomial(j, n - k)
%o A226157            for j in (0..n))/((-m)^k*(k+1)) for k in (0..n)))
%o A226157 def A226157(n):   # The numerators of BS(n, 2) relative to A225481
%o A226157     C = mul(filter(lambda p: ((n+1)%p == 0) or (n%(p-1) == 0), primes(n+2)))
%o A226157     return C*BS(n, 2)
%o A226157 [A226157(n) for n in (0..25)]
%Y A226157 Cf. A225481, A226156.
%K A226157 sign,frac
%O A226157 0,3
%A A226157 _Peter Luschny_, May 30 2013