A346714 - OEIS

%I #8 Aug 01 2021 12:57:33

%S 0,2,2,-4,-26,100,74,-964,-7438,45972,486806,-218644,-55169462,

%T 662381044,328098718,-21019858108,-9375927526,20435090284868,

%U 1203410229242,-1072522899634372,-13371923279768194,93925510336152268,4777233165759979134,-179655667413148948

%N The numerators of the semiderivative of the Euler polynomials at x = 1 and normalized by sqrt(Pi).

%C The semiderivative is the fractional derivative of order 1/2. The Davison-Essex method is used.

%D M. Davison and C. Essex, Fractional differential equations and initial value problems, The Mathematical Scientist, vol. 23, no. 2, pp. 108-116, 1998.

%F r(n) = Integral_{0..1}((d/dx)Euler(n, x) / sqrt(1 - x)).

%F a(n) = numerator(r(n)).

%e r(n) = 0, 2, 2/3, -4/5, -26/35, 100/63, 74/33, -964/143, -7438/585, 45972/935, ...

%e a(n) = numerator(sde_n(1) - sde_n(0)), where

%e sde_0(x) =  0

%e sde_1(x) = -2*sqrt(1-x);

%e sde_2(x) = (-2 - 4*x)*sqrt(1-x) / 3;

%e sde_3(x) = (4 + 2*x - 6*x^2)*sqrt(1-x) / 5;

%e sde_4(x) = (26 + 48*x + 36*x^2 - 40*x^3)*sqrt(1-x) / 35;

%e sde_5(x) = (-100 - 50*x + 120*x^2 + 100*x^3 - 70*x^4)*sqrt(1-x) / 63;

%e sde_6(x) = (-74 - 136*x - 102*x^2 + 80*x^3 + 70*x^4 - 36*x^5)*sqrt(1-x) / 33.

%p r := n -> int(diff(euler(n, x), x) / sqrt(1 - x), x = 0..1);

%p a := n -> numer(r(n)): seq(a(n), n=0..23);

%p # Alternative:

%p fe := n -> sqrt(Pi)*fracdiff(euler(n, x), x, 1/2):

%p seq(numer(simplify(subs(x=1, fe(n)))), n = 0..9);

%Y Cf. A346709, A346710, A346711, A346712, A346715 (denominator).

%K sign,frac

%O 0,2

%A _Peter Luschny_, Jul 31 2021