Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #11 May 15 2017 18:25:11
%S 1,-1,-1,3,7,-25,-31,427,127,-12465,-2555,555731,1414477,-35135945,
%T -57337,2990414715,118518239,-329655706465,-5749691557,45692713833379,
%U 91546277357,-7777794952988025,-1792042792463,1595024111042171723,1982765468311237,-387863354088927172625
%N Numerator of Bernoulli(n, 1/4).
%C From _Wolfdieter Lang_, Apr 28 2017: (Start)
%C The rationals r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A285061(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) define generalized Bernoulli numbers, named B[4,1](n), in terms of the generalized Stirling2 numbers S2[4,1]. The numerators of r(n) are a(n) and the denominators A141459(n). r(n) = B[4,1](n) = 4^n*B(n, 1/4) with the Bernoulli polynomials B(n, x) = Bernoulli(n, x) from A196838/A196839 or A053382/A053383.
%C The generalized Bernoulli numbers B[4,3](n) = Sum_{k=0..n} ((-1)^k/(k+1))* A225467(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) satisfy
%C B[4,3](n) = 4^n*B(n, 3/4) = (-1)^n*B[4,1](n). They have numerators (-1)^n*a(n) and also denominators A141459(n). (End)
%H Vincenzo Librandi, <a href="/A157817/b157817.txt">Table of n, a(n) for n = 0..250</a>
%F From _Wolfdieter Lang_, Apr 28 2017: (Start)
%F a(n) = numerator(Bernoulli(n, 1/4)) with denominator A157818(n) (see the name).
%F a(n) = numerator(4^n*Bernoulli(n, 1/4)) with denominator A141459(n) = A157818(n)/4^n.
%F a(n)*(-1)^n = numerator(4^n*Bernoulli(n, 3/4)) with denominator A141459(n).
%F (End)
%t Table[Numerator[BernoulliB[n, 1/4]], {n, 0, 50}] (* _Vincenzo Librandi_, Mar 16 2014 *)
%Y For denominators see A157818 and A141459.
%K sign,easy,frac
%O 0,4
%A _N. J. A. Sloane_, Nov 08 2009