A346711 Numerators of a fractional order v differentiation of the Bernoulli polynomials with v = 1/2, evaluated at x = 1 and normalized by sqrt(Pi).

1, 3, 5, 1, -23, -5, 5, 521, -1357, -97, 35713, 538019, -45411, -109923, 2173451, 12637, -3109585853579, -2750583611, 16296301543, 41079818933, -154715264921, -2559782104871, 201299334909241, 8079972368723417, -2104258043122757, -118316122614712593, 418629788956582261

Offset

0,2

Comments

a(n) = numerator(r(n)). Here r(n) = Pi^(1/2)*D^(1/2)(B(n, x))|x=1, where D^v denotes a fractional differentiation operator of order v and f(x)|x=k denotes the evaluation of f(x) at k. B(n, x) are the Bernoulli polynomials. The operator D^v is defined by linear extension of D^(v)(x^n) = (Gamma(n + 1)/Gamma(n + 1 - v)) * x^(n - v) to polynomials.

A more sophisticated definition of a semiderivative of the Bernoulli polynomials is in A346709.

Links

Table of n, a(n) for n=0..26.

Example

r(n) = 1, 3/2, 5/6, 1/5, -23/210, -5/63, 5/66, 521/6435, -1357/12870, -97/663, 35713/149226, ...
a(n) = numerator(sds_n(1)), where
sds_0(x) = 1/x^(1/2);
sds_1(x) = (1/2)*(-1 + 4*x)/x^(1/2);
sds_2(x) = (1/6)*(1 - 12*x + 16*x^2)/x^(1/2);
sds_3(x) = (1/5)*(5 - 20*x + 16*x^2)*x^(1/2);
sds_4(x) = (1/210)*(-7 + 560*x^2 - 1344*x^3 + 768*x^4)/x^(1/2);
sds_5(x) = (1/63)*(-21 + 336*x^2 - 576*x^3 + 256*x^4)*x^(1/2);
sds_6(x) = (1/462)*(11 - 616*x^2 + 4224*x^4 - 5632*x^5 + 2048*x^6)/x^(1/2).

Maple

A346711frac := proc(n) local der, ext, p, v;
der := (d, n) -> (GAMMA(n+1)/GAMMA(n+1-d))*x^(n-d):
ext := (d, p) -> add(coeff(p, x, k)*der(d, k), k=min(floor(d), 1)..degree(p)):
p := ext(1/2, bernoulli(n, x)):
v := sqrt(Pi)*subs(x=1, p) end:
a := n -> numer(A346711frac(n)):
seq(a(n), n=0..26);

Crossrefs

A346712 (denominator), A346709.

Sequence in context: A265649 A216520 A204161 * A278968 A220110 A327693

Adjacent sequences: A346708 A346709 A346710 * A346712 A346713 A346714

Keyword

sign,frac

Author

Peter Luschny, Jul 30 2021

Status

approved