A346714 The numerators of the semiderivative of the Euler polynomials at x = 1 and normalized by sqrt(Pi).

0, 2, 2, -4, -26, 100, 74, -964, -7438, 45972, 486806, -218644, -55169462, 662381044, 328098718, -21019858108, -9375927526, 20435090284868, 1203410229242, -1072522899634372, -13371923279768194, 93925510336152268, 4777233165759979134, -179655667413148948

Offset

0,2

Comments

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

References

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

Links

Table of n, a(n) for n=0..23.

Formula

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

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

Example

r(n) = 0, 2, 2/3, -4/5, -26/35, 100/63, 74/33, -964/143, -7438/585, 45972/935, ...
a(n) = numerator(sde_n(1) - sde_n(0)), where
sde_0(x) =  0
sde_1(x) = -2*sqrt(1-x);
sde_2(x) = (-2 - 4*x)*sqrt(1-x) / 3;
sde_3(x) = (4 + 2*x - 6*x^2)*sqrt(1-x) / 5;
sde_4(x) = (26 + 48*x + 36*x^2 - 40*x^3)*sqrt(1-x) / 35;
sde_5(x) = (-100 - 50*x + 120*x^2 + 100*x^3 - 70*x^4)*sqrt(1-x) / 63;
sde_6(x) = (-74 - 136*x - 102*x^2 + 80*x^3 + 70*x^4 - 36*x^5)*sqrt(1-x) / 33.

Maple

r := n -> int(diff(euler(n, x), x) / sqrt(1 - x), x = 0..1);
a := n -> numer(r(n)): seq(a(n), n=0..23);
# Alternative:
fe := n -> sqrt(Pi)*fracdiff(euler(n, x), x, 1/2):
seq(numer(simplify(subs(x=1, fe(n)))), n = 0..9);

Crossrefs

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

Sequence in context: A006829 A154594 A098335 * A049147 A189879 A189870

Adjacent sequences: A346711 A346712 A346713 * A346715 A346716 A346717

Keyword

sign,frac

Author

Peter Luschny, Jul 31 2021

Status

approved