A239792 Numerator of b_{2n}(1/4), where b_{n}(x) are Nörlund's generalized Bernoulli polynomials.

1, -1, 3, -61, 1261, -4977, 999645, -16820653, 288427601, -1975649524361, 250373334235999, -741069328361243, 2017175162278526957, -16484758150014378103, 1866091048556360006871, -747145289541069391049541, 558035966935526487401599645, -94004035636878314426017611

Offset: 0

Author

Peter Luschny, Mar 26 2014

Keywords

• sign
• frac

References

• Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969, page 34.
• N. E. Nörlund, Vorlesungen über Differenzenrechnung, Berlin, 1924.

Links

• J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. 15 (1) (1966), 43-45.

Crossrefs

Cf. A220412, A239793 (denominators).

Programs

Maple

b := proc(n) option remember; if n < 1 then 1 else
-add(binomial(n-1, k-1)*bernoulli(k)*b(n-k)/k, k= 2..n)/2 fi end:
A239792 := n -> numer(b(2*n));
seq(A239792(n), n=0..17);

Mathematica

b[n_] := b[n] = If[n < 1, 1, -Sum[Binomial[n - 1, k - 1] BernoulliB[k] b[n - k]/k, {k, 2, n}]/2];
a[n_] := b[2n] // Numerator;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 28 2019, from Maple *)

Formula

Let b(n) = -Sum_{k=2..n} (C(n-1, k-1)*Bernoulli(k)*b(n-k)/k)/2 for n>0 and otherwise 1. Then a(n) = numerator(b(2*n)).