The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 2
 1, -3, 65, -7, -166311, -3326411, 250810573, -15, -70140643372783, -16050395192832019, 1253057168563221, 489854682254665727977, -4242091290877439975, -567128617209289175040000027, -469414018487906631382763519971, -31, -99189110152385088675839967, 60136002178464962241806622916607999965, 655685669998967370706944000037, -195445976621261878742262620176483614720000039 (list; graph; refs; listen; history; text; internal format)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 3 13:56 EST 2022. Contains 358534 sequences. (Running on oeis4.)