A239793 -id:A239793 - cp's OEIS frontend

A239795 a(n) = A239793(n)/2^(3*n).

1, 3, 5, 21, 45, 11, 91, 45, 17, 1995, 3465, 115, 2925, 189, 145, 341, 1309, 1, 9139, 65, 2255, 148995, 108675, 1645, 270725, 21879, 583, 4389, 4959, 59, 1548729, 27027, 60775, 130985, 15525, 1065, 66047553, 2567565, 39, 2133, 56457, 1411, 8161615, 2639

Offset: 0

Peter Luschny, Mar 26 2014

nonn

See A239792 for references.

Cf. A220412, A239792.

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:
A239795 := n -> denom(b(2*n))/2^(3*n):
seq(A239795(n), n=0..43);

Let b(n) = -Sum_{2<=k<=n} (C(n-1, k-1)*Bernoulli(k)*b(n-k)/k)/2

for n>0 and otherwise 1. Then a(n) = denominator(b(2*n))/2^(3*n).

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

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Mar 26 2014

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

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

Cf. A220412, A239793 (denominators).

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);

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 *)

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)).

cp's OEIS Frontend

A239795 a(n) = A239793(n)/2^(3*n).

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