A324303 a(n) = [x^(n*(n+2))] Sum_{m>=0} x^m * (x^(m+1) + i)^m / (1 + i*x^(m+1))^(m+1), for n >= 0.

1, 3, 15, 33, 323, 1753, 8991, 58495, 268541, 1462761, 8097455, 13837475, 251595971, 1409934009, 9674895101, 46242140669, 252055236611, 1426069096367, 8079317057871, 45129781821155, 260949012214457, 1482376214228889, 8443414161166175, 28159474726319741, 274738209468977925, 1569245074591691433, 8970254867527278283, 51182160225347277091, 293733710358893793731, 1737504244377865756267

Offset

0,2

Comments

a(n) = A324300(n*(n+2)) for n >= 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..179

Formula

a(n) = [x^(n*(n+2))] Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1).

a(n) = [x^(n*(n+2))] Sum_{n>=0} x^n * (x^(n+1) - i)^n / (1 - i*x^(n+1))^(n+1).

a(n) = [x^(n*(n+2))] Sum_{n>=0} (i*x)^n * (1 - i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).

a(n) = [x^(n*(n+2))] Sum_{n>=0} (-i*x)^n * (1 + i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).

Example

The g.f. of A324300 is given by
G(x) = Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1)
where
G(x) = 1 - 2*x^2 + 3*x^3 + 2*x^4 - 2*x^6 - 14*x^7 + 15*x^8 - 2*x^10 + 22*x^11 + 2*x^12 - 84*x^14 + 33*x^15 + 2*x^16 - 2*x^18 + 38*x^19 + 172*x^20 - 2*x^22 - 508*x^23 + 323*x^24 - 292*x^26 + 54*x^27 + 2*x^28 - 2*x^30 + 1088*x^31 + 444*x^32 - 2580*x^34 + 1753*x^35 + ...
This sequence gives the odd coefficients of x^n in G(x), which occur at n = k*(k+2) for k >= 0.
Explicitly,
G(x) = 1/(1+i*x) + x*(x^2+i)/(1+i*x^2)^2 + x^2*(x^3+i)^2/(1+i*x^3)^3 + x^3*(x^4+i)^3/(1+i*x^4)^4 + x^4*(x^5+i)^4/(1+i*x^5)^5 + x^5*(x^6+i)^5/(1+i*x^6)^6 + x^6*(x^7+i)^6/(1+i*x^7)^7 + x^7*(x^8+i)^7/(1+i*x^8)^8 + ...
also
G(x) = (1-i*x)/(1+x^2) + i*x*(1-i*x^2)^3/(1+x^4)^2 - x^2*(1-i*x^3)^5/(1+x^6)^3 - i*x^3*(1-i*x^4)^7/(1+x^8)^4 + x^4*(1-i*x^5)^9/(1+x^10)^5 + i*x^5*(1-i*x^6)^11/(1+x^12)^6 + i*x^6*(1-i*x^7)^11/(1+x^14)^7 + ...
Note that the imaginary components in the above sums vanish.

Prog

(PARI) {A324300(n) = my(SUM = sum(m=0, n, x^m*(x^(m+1) + I +x*O(x^n))^m / (1 + I*x^(m+1) +x*O(x^n))^(m+1) ) ); polcoeff(SUM, n)}
{a(n) = A324300(n*(n+2))}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A324300, A324301, A324302.

Cf. A323687 (variant).

Sequence in context: A031091 A211441 A086380 * A009261 A372735 A331249

Adjacent sequences: A324300 A324301 A324302 * A324304 A324305 A324306

Keyword

nonn

Author

Paul D. Hanna, Feb 21 2019

Status

approved