A326429 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.

1, -3, 65, -7, -166311, -3326411, 250810573, -15, -70140643372783, -16050395192832019, 1253057168563221, 489854682254665727977, -4242091290877439975, -567128617209289175040000027, -469414018487906631382763519971, -31, -99189110152385088675839967, 60136002178464962241806622916607999965, 655685669998967370706944000037, -195445976621261878742262620176483614720000039

Offset

1,2

Comments

More generally, the following sums are equal:

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

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

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

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

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

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

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

Example

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! + ...
such that
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! + ...
also
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! + ...
COEFFICIENTS OF x^(2^n) IN A(x).
The coefficients of x^(2^n)/(2^n)! in e.g.f. A(x), starting with n = 1, begin:
[1, -3, -7, -15, -31, -63, -127, -255, -511, -1023, -2047, ...].
RELATED POLYNOMIALS.
The polynomials real( (x^n + i)^n ) begin:
n=0: 1,
n=1: x,
n=2: x^4 - 1,
n=3: x^9 - 3*x^3,
n=4: x^16 - 6*x^8 + 1,
n=5: x^25 - 10*x^15 + 5*x^5,
n=6: x^36 - 15*x^24 + 15*x^12 - 1,
n=7: x^49 - 21*x^35 + 35*x^21 - 7*x^7,
n=8: x^64 - 28*x^48 + 70*x^32 - 28*x^16 + 1,
...
The polynomials imag( (x^n + i)^n ) begin:
n=0: 0,
n=1: 1,
n=2: 2*x^2,
n=3: 3*x^6 - 1,
n=4: 4*x^12 - 4*x^4,
n=5: 5*x^20 - 10*x^10 + 1,
n=6: 6*x^30 - 20*x^18 + 6*x^6,
n=7: 7*x^42 - 35*x^28 + 21*x^14 - 1,
n=8: 8*x^56 - 56*x^40 + 56*x^24 - 8*x^8,
...
RELATED SERIES.
At x = 1/2, we have
A(1/2) = sin(-1) + Sum_{n>=0} sin(1/2^n) * real( (1/2^n + i)^n ) / n!, also,
A(1/2) = sin(-1) + Sum_{n>=0} cos(1/2^n) * imag( (1/2^n + i)^n ) / n!,
where A(1/2) = 0.11855108754295937931093066450327494094096154528452247568757943...

Prog

(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)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A326425.

Sequence in context: A304288 A275044 A205645 * A300010 A196798 A196586

Adjacent sequences: A326426 A326427 A326428 * A326430 A326431 A326432

Keyword

sign

Author

Paul D. Hanna, Jul 04 2019

Status

approved