A389357 - OEIS

%I #11 Nov 16 2025 13:37:30

%S 1,7,149,2161,53089,1187803,62566171,126420629,34676275663,

%T 1327006348769,762860964413,35274884855393,1417459086247517,

%U 8539282786885727,994152675906323707,61839074498178353459,1984721470514398782313,27861319566938771335907,4133713434815611021027361,8286341865806624403560347

%N Numerators E(n) in the closed form of the integral J(2n+1) = Integral_{x=0..Pi/2} x * cos(x)^(2n+1) dx = binomial(2n+1,n)/2^(2n+2) * Pi - E(n)/F(n).

%C Appears in the exact evaluation of integrals with odd powers of cosine.

%C Companion denominators are sequence F(n).

%C For example, J(5) = 252/2^13 * Pi - 21/100 so E(5)=21, F(5)=100.

%F a(n) = numerator(R(n)) where R(n) = E(n)/F(n) = (1/2^(2*n)) * Sum_{1 <= j <= 2n+1, j odd} binomial(2*n+1,(2*n+1-j)/2)/j^2.

%F J(2*n+1) = (2*n)!!/(2*n+1)!! * Pi/2 - R(n).

%F Proof for odd powers J(2n+1) = Integral_{0..Pi/2} x*cos(x)^(2n+1) dx.

%F 1) Expand cos(x)^(2n+1) = ( (e^{ix}+e^{-ix})/2 )^(2n+1); only odd harmonics survive.

%F 2) Integrate termwise: Integral_{0..Pi/2} x*cos(jx) dx = (Pi/2)*sin(jPi/2)/j - 1/j^2, valid for odd j.

%F 3) Summing gives J(2n+1) = (2n)!!/(2n+1)!! * Pi/2 - (1/2^(2n)) * Sum_{j odd<=2n+1} binomial(2n+1,(2n+1-j)/2)/j^2.

%F 4) Writing the rational sum in lowest terms as E(n)/F(n) with gcd(E,F)=1 yields J(2n+1) = (2n)!!/(2n+1)!! * Pi/2 - E(n)/F(n).

%e n=3: J(7) = 8/35 * Pi - 2161/3675, so E(3)=2161, F(3)=3675.

%o (Python)

%o # generates sequence and numerical validation between the numerical integration using high-precision quadrature and the closed form formula.

%o import mpmath as mp

%o from math import comb

%o from fractions import Fraction

%o mp.mp.dps = 80

%o eps = mp.mpf('1e-12')

%o def factorial2(n):

%o     if n <= 0: return 1

%o     res = 1

%o     for k in range(n,0,-2): res *= k

%o     return mp.mpf(res)

%o def R_frac(n):

%o     s = Fraction(0,1)

%o     for j in range(1,2*n+2,2):  # odd j

%o         s += Fraction(comb(2*n+1,(2*n+1-j)//2), j*j)

%o     return s / 2**(2*n)

%o def E(n): return R_frac(n).numerator

%o def F(n): return R_frac(n).denominator

%o # Numeric integral

%o def J_numeric(n):

%o     f = lambda x: x * mp.cos(x)**(2*n+1)

%o     return mp.quad(f, [0, mp.pi/2])

%o # Closed form

%o def J_formula(n):

%o     coeff = factorial2(2*n) / factorial2(2*n+1) * mp.pi/2

%o     return coeff - mp.mpf(E(n)) / mp.mpf(F(n))

%o print([E(k) for k in range(1,20)])

%o for n in [1,2,3,5,7]:

%o     val_num = J_numeric(n)

%o     val_form = J_formula(n)

%o     err = abs(val_num - val_form)

%o     match = err < 10*eps

%o     print(f"n={n}: match={match}, error={err}")

%Y Cf. A000984 (central binomial coefficients), A389342 and A387583 for sequences C(n) and D(n) for even case, F(n) denominators for odd case.

%K nonn,frac

%O 0,2

%A _Patrick Demichel_, Oct 01 2025