A323681 G.f.: Sum_{n>=0} x^n*((1+x)^n + i)^n / (1 + i*x*(1+x)^n)^(n+1), where i^2 = -1.

1, 1, 2, 7, 22, 87, 377, 1771, 9026, 49199, 284983, 1745336, 11246563, 75956728, 535909242, 3938660615, 30078439304, 238154159543, 1951238032473, 16514089454284, 144148618179948, 1295871420550063, 11982543274136961, 113830968212019730, 1109755421437926323, 11092205946446644962, 113562177701272805808, 1189885690276586123039, 12749384941695403919951, 139593699183914764551501, 1560760177586802637547293

Offset

0,3

Comments

Note that the generating function expands to a power series in x consisting of only real coefficients.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f.: Sum_{n>=0} x^n*((1+x)^n + i)^n / (1 + i*x*(1+x)^n)^(n+1).

G.f.: Sum_{n>=0} x^n*((1+x)^n - i)^n / (1 - i*x*(1+x)^n)^(n+1).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 22*x^4 + 87*x^5 + 377*x^6 + 1771*x^7 + 9026*x^8 + 49199*x^9 + 284983*x^10 + 1745336*x^11 + 11246563*x^12 + ...
such that
A(x) = 1/(1 + i*x) + x*((1+x) + i)/(1 + i*x*(1+x))^2 + x^2*((1+x)^2 + i)^2/(1 + i*x*(1+x)^2)^3 + x^3*((1+x)^3 + i)^3/(1 + i*x*(1+x)^3)^4 + x^4*((1+x)^4 + i)^4/(1 + i*x*(1+x)^4)^5 + x^5*((1+x)^5 + i)^5/(1 + i*x*(1+x)^5)^6 + x^6*((1+x)^6 + i)^6/(1 + i*x*(1+x)^6)^7 + x^7*((1+x)^7 + i)^7/(1 + i*x*(1+x)^7)^8 + ...
also,
A(x) = 1/(1 - i*x) + x*((1+x) - i)/(1 - i*x*(1+x))^2 + x^2*((1+x)^2 - i)^2/(1 - i*x*(1+x)^2)^3 + x^3*((1+x)^3 - i)^3/(1 - i*x*(1+x)^3)^4 + x^4*((1+x)^4 - i)^4/(1 - i*x*(1+x)^4)^5 + x^5*((1+x)^5 - i)^5/(1 - i*x*(1+x)^5)^6 + x^6*((1+x)^6 - i)^6/(1 - i*x*(1+x)^6)^7 + x^7*((1+x)^7 - i)^7/(1 - i*x*(1+x)^7)^8 + ...
RELATED INFINITE SERIES.
At x = -1/2, the g.f. A(x=-1/2) diverges, but the related series converges:
S = Sum_{n>=0} (-1/2)^n * (1/2^n + i)^n / (1 - i/2^(n+1))^(n+1).
Equivalently,
S = Sum_{n>=0} (-1)^n * 2^(n+1) * (1 + 2^n*i)^n / (2^(n+1) - i)^(n+1) ;
written explicitly,
S = 2/(2-i) - 2^2*(1+2*i)/(2^2-i)^2 + 2^3*(1+2^2*i)^2/(2^3-i)^3
- 2^4*(1+2^3*i)^3/(2^4-i)^4 + 2^5*(1+2^4*i)^4/(2^5-i)^5
- 2^6*(1+2^5*i)^5/(2^6-i)^6 + 2^7*(1+2^6*i)^6/(2^7-i)^7 + ...
which equals the real number
S = 0.61999741931719746274134412657304059740143377356135821449819330...

Prog

(PARI) {a(n) = my(A = sum(m=0, n+1, x^m*((1+x +x*O(x^n) )^m + I)^m/(1 + I*x*(1+x +x*O(x^n) )^m )^(m+1) )); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n+1, x^m*((1+x +x*O(x^n) )^m - I)^m/(1 - I*x*(1+x +x*O(x^n) )^m )^(m+1) )); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

Crossrefs

Cf. A323680, A323682, A323683, A323684, A323685.

Cf. A323570.

Sequence in context: A150327 A150328 A150329 * A150330 A150331 A150332

Adjacent sequences: A323678 A323679 A323680 * A323682 A323683 A323684

Keyword

nonn

Author

Paul D. Hanna, Feb 11 2019

Status

approved