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A326425
E.g.f.: 2*cos(1) - Sum_{n>=0} (x^n + i)^n * exp(-i*x^n) / n!, where i^2 = -1.
5
1, -2, 0, 8, -42, -12, 0, 16, 35280, -342740, 0, 24, -28607040, -28, 0, 494010316832, -7553895148800, -36, 0, 40, -74419125580800, -1209692942954496044, 0, 48, 75042863061697398067200, -1665300744929840578560052, 0, 7481696749074925977600056, -737929590355077120000, -60, 0, 64, -1845244146938745498017569136640000, -98180983578103240698416332800068, 0, 316965213413352778993346090631168000072
OFFSET
0,2
COMMENTS
More generally, the following sums are equal:
(1) 2*cosh(p*r) - Sum_{n>=0} (q^n + p)^n * exp(-p*q^n*r) * r^n/n!,
(2) 2*cosh(p*r) - Sum_{n>=0} (q^n - p)^n * exp(+p*q^n*r) * r^n/n!,
under suitable conditions; here, p = sqrt(-1), q = x, r = 1.
LINKS
FORMULA
E.g.f.: 2*cos(1) - Sum_{n>=0} (x^n + i)^n * exp(-i*x^n) / n!.
E.g.f.: 2*cos(1) - Sum_{n>=0} (x^n - i)^n * exp(+i*x^n) / n!.
E.g.f.: 2*cos(1) - Sum_{n>=0} cos(x^n) * real((x^n + i)^n) / n! - Sum_{n>=0} sin(x^n) * imag((x^n + i)^n) / n!.
a(4*n+2) = 0 for n >= 0.
EXAMPLE
E.g.f.: A(x) = 1 - 2*x + 8*x^3/3! - 42*x^4/4! - 12*x^5/5! + 16*x^7/7! + 35280*x^8/8! - 342740*x^9/9! + 24*x^11/11! - 28607040*x^12/12! - 28*x^13/13! + 494010316832*x^15/15! - 7553895148800*x^16/16! - 36*x^17/17! + 40*x^19/19! - 74419125580800*x^20/20! - 1209692942954496044*x^21/21! + 48*x^23/23! + 75042863061697398067200*x^24/24! - 1665300744929840578560052*x^25/25! + ...
such that
A(x) = 2*cos(1) - (exp(-i) + (x+i)*exp(-i*x) + (x^2+i)^2*exp(-i*x^2)/2! + (x^3+i)^3*exp(-i*x^3)/3! + (x^4+i)^4*exp(-i*x^4)/4! + (x^5+i)^5*exp(-i*x^5)/5! + (x^6+i)^6*exp(-i*x^6)/6! + (x^7+i)^7*exp(-i*x^7)/7! + ...)
in which the imaginary components vanish upon expanding into a power series in x.
RELATED SERIES.
2*cos(1) - Sum_{n>=0} cos(x^n) * real((x^n + i)^n) / n! = 1 - x + 6*x^3/3! - 18*x^4/4! - 10*x^5/5! + 14*x^7/7! + 21840*x^8/8! - 151218*x^9/9! + 22*x^11/11! - 20623680*x^12/12! - 26*x^13/13! + 272432160030*x^15/15! + ...
Sum_{n>=0} sin(x^n) * imag((x^n + i)^n) / n! = x - 2*x^3/3! + 24*x^4/4! + 2*x^5/5! - 2*x^7/7! - 13440*x^8/8! + 191522*x^9/9! - 2*x^11/11! + 7983360*x^12/12! + 2*x^13/13! - 221578156802*x^15/15! + ...
A related identity is that the following series are equal:
S(x) = Sum_{n>=0} cos(x^n) * imag( (x^n + i)^n ) / n!,
S(x) = Sum_{n>=0} sin(x^n) * real( (x^n + i)^n ) / n!,
where
S(x) = sin(1) + 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! + ... + A326429(n)*x^(2*n)/(2*n)! + ...
Another related identity:
tan(b - Pi) = (Sum_{n>=0} imag((a + b*i)^n)/n!) / (Sum_{n>=0} real((a + b*i)^n)/n!), and holds for all real valued a and b.
PROG
(PARI) /* quick informal code for the initial 120 terms */
\p500
Vec(round(serlaplace( 2*cos(1) - sum(n=0, 500, (x^n + I +O(x^121))^n * exp(-I*x^n +O(x^121)) * 1./n! ) )))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 03 2019
STATUS
approved