A326429 - OEIS

%I #29 Jul 12 2019 17:43:14

%S 1,-3,65,-7,-166311,-3326411,250810573,-15,-70140643372783,

%T -16050395192832019,1253057168563221,489854682254665727977,

%U -4242091290877439975,-567128617209289175040000027,-469414018487906631382763519971,-31,-99189110152385088675839967,60136002178464962241806622916607999965,655685669998967370706944000037,-195445976621261878742262620176483614720000039

%N E.g.f.: A(x) = sin(-1) + Sum_{n>=0} sin(x^n) * real( (x^n + i)^n ) / n!, an even function, showing only the coefficients of x^(2*n)/(2*n)! in A(x) for n >= 1.

%C More generally, the following sums are equal:

%C (1) sinh(-p*r) + Sum_{n>=0} sinh(p*q^n*r) * ((q^n + p)^n + (q^n - p)^n)/2 * r^n/n!,

%C (2) sinh(-p*r) + Sum_{n>=0} cosh(p*q^n*r) * ((q^n + p)^n - (q^n - p)^n)/2 * r^n/n!,

%C under suitable conditions; here, p = i = sqrt(-1), q = x, r = 1.

%C What is the radius of convergence of the e.g.f. A(x) when expanded as a power series in x?

%H Paul D. Hanna, <a href="/A326429/b326429.txt">Table of n, a(n) for n = 1..520</a>

%F E.g.f. A(x) = Sum_{n>=1} a(n)*x^(2*n)/(2*n)! equals the following sums.

%F E.g.f.: sin(-1) + Sum_{n>=0} sin(x^n) * real( (x^n + i)^n ) / n!.

%F E.g.f.: sin(-1) + Sum_{n>=0} cos(x^n) * imag( (x^n + i)^n ) / n!.

%F a(2^n) = 1 - 2^(n+1) for n >= 1.

%e E.g.f.: A(x) = x^2/2! - 3*x^4/4! + 65*x^6/6! - 7*x^8/8! - 166311*x^10/10! - 3326411*x^12/12! + 250810573*x^14/14! - 15*x^16/16! - 70140643372783*x^18/18! - 16050395192832019*x^20/20! + ...

%e such that

%e A(x) = sin(-1) + sin(1)*(1) + sin(x)*(x) + sin(x^2)*(x^4 - 1)/2! + sin(x^3)*(x^9 - 3*x^3)/3! + sin(x^4)*(x^16 - 6*x^8 + 1)/4! + sin(x^5)*(x^25 - 10*x^15 + 5*x^5)/5! + sin(x^6)*(x^36 - 15*x^24 + 15*x^12 - 1)/6! + sin(x^7)*(x^49 - 21*x^35 + 35*x^21 - 7*x^7)/7! + sin(x^8)*(x^64 - 28*x^48 + 70*x^32 - 28*x^16 + 1)/8! + ...

%e also

%e A(x) = sin(-1) + cos(1)*(0) + cos(x)*(1) + cos(x^2)*(2*x^2)/2! + cos(x^3)*(3*x^6 - 1)/3! + cos(x^4)*(4*x^12 - 4*x^4)/4! + cos(x^5)*(5*x^20 - 10*x^10 + 1)/5! + cos(x^6)*(6*x^30 - 20*x^18 + 6*x^6)/6! + cos(x^7)*(7*x^42 - 35*x^28 + 21*x^14 - 1)/7! + cos(x^8)*(8*x^56 - 56*x^40 + 56*x^24 - 8*x^8)/8! + ...

%e COEFFICIENTS OF x^(2^n) IN A(x).

%e The coefficients of x^(2^n)/(2^n)! in e.g.f. A(x), starting with n = 1, begin:

%e [1, -3, -7, -15, -31, -63, -127, -255, -511, -1023, -2047, ...].

%e RELATED POLYNOMIALS.

%e The polynomials real( (x^n + i)^n ) begin:

%e n=0: 1,

%e n=1: x,

%e n=2: x^4 - 1,

%e n=3: x^9 - 3*x^3,

%e n=4: x^16 - 6*x^8 + 1,

%e n=5: x^25 - 10*x^15 + 5*x^5,

%e n=6: x^36 - 15*x^24 + 15*x^12 - 1,

%e n=7: x^49 - 21*x^35 + 35*x^21 - 7*x^7,

%e n=8: x^64 - 28*x^48 + 70*x^32 - 28*x^16 + 1,

%e ...

%e The polynomials imag( (x^n + i)^n ) begin:

%e n=0: 0,

%e n=1: 1,

%e n=2: 2*x^2,

%e n=3: 3*x^6 - 1,

%e n=4: 4*x^12 - 4*x^4,

%e n=5: 5*x^20 - 10*x^10 + 1,

%e n=6: 6*x^30 - 20*x^18 + 6*x^6,

%e n=7: 7*x^42 - 35*x^28 + 21*x^14 - 1,

%e n=8: 8*x^56 - 56*x^40 + 56*x^24 - 8*x^8,

%e ...

%e RELATED SERIES.

%e At x = 1/2, we have

%e A(1/2) = sin(-1) + Sum_{n>=0} sin(1/2^n) * real( (1/2^n + i)^n ) / n!, also,

%e A(1/2) = sin(-1) + Sum_{n>=0} cos(1/2^n) * imag( (1/2^n + i)^n ) / n!,

%e where A(1/2) = 0.11855108754295937931093066450327494094096154528452247568757943...

%o (PARI) {a(n) = my(A = sum(m=1,2*n+1, sin(x^m +x*O(x^(2*n))) * real( (x^m + I)^m ) / m! )); (2*n)!*polcoeff(A,2*n)}

%o for(n=1,25,print1(a(n),", "))

%Y Cf. A326425.

%K sign

%O 1,2

%A _Paul D. Hanna_, Jul 04 2019