A346711 - OEIS

%I #12 Aug 01 2021 12:57:24

%S 1,3,5,1,-23,-5,5,521,-1357,-97,35713,538019,-45411,-109923,2173451,

%T 12637,-3109585853579,-2750583611,16296301543,41079818933,

%U -154715264921,-2559782104871,201299334909241,8079972368723417,-2104258043122757,-118316122614712593,418629788956582261

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

%C 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.

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

%e r(n) = 1, 3/2, 5/6, 1/5, -23/210, -5/63, 5/66, 521/6435, -1357/12870, -97/663, 35713/149226, ...

%e a(n) = numerator(sds_n(1)), where

%e sds_0(x) = 1/x^(1/2);

%e sds_1(x) = (1/2)*(-1 + 4*x)/x^(1/2);

%e sds_2(x) = (1/6)*(1 - 12*x + 16*x^2)/x^(1/2);

%e sds_3(x) = (1/5)*(5 - 20*x + 16*x^2)*x^(1/2);

%e sds_4(x) = (1/210)*(-7 + 560*x^2 - 1344*x^3 + 768*x^4)/x^(1/2);

%e sds_5(x) = (1/63)*(-21 + 336*x^2 - 576*x^3 + 256*x^4)*x^(1/2);

%e sds_6(x) = (1/462)*(11 - 616*x^2 + 4224*x^4 - 5632*x^5 + 2048*x^6)/x^(1/2).

%p A346711frac := proc(n) local der, ext, p, v;

%p der := (d, n) -> (GAMMA(n+1)/GAMMA(n+1-d))*x^(n-d):

%p ext := (d, p) -> add(coeff(p,x,k)*der(d, k), k=min(floor(d),1)..degree(p)):

%p p := ext(1/2, bernoulli(n, x)):

%p v := sqrt(Pi)*subs(x=1, p) end:

%p a := n -> numer(A346711frac(n)):

%p seq(a(n), n=0..26);

%Y A346712 (denominator), A346709.

%K sign,frac

%O 0,2

%A _Peter Luschny_, Jul 30 2021