A323684 G.f.: Sum_{n>=0} x^n * ((1+x)^n + 2*i)^n / (1 + 2*i*x*(1+x)^n)^(n+1), where i^2 = -1.

1, 1, 2, 13, 40, 177, 959, 4927, 28118, 173747, 1116769, 7572422, 53958439, 400843780, 3100289406, 24912295541, 207402869744, 1785581874833, 15870223469439, 145385654206288, 1370808547372554, 13286036772531961, 132211223720926353, 1349366082883824538, 14110857413371538549, 151058713428413686374, 1654028328360981132108

Offset

0,3

Comments

Note that the generating function expands into a power series in x with only real coefficients.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

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

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 13*x^3 + 40*x^4 + 177*x^5 + 959*x^6 + 4927*x^7 + 28118*x^8 + 173747*x^9 + 1116769*x^10 + 7572422*x^11 + ...
Let r = 2*i, so that r^2 = -4, then
A(x) = 1/(1 + r*x) + x*((1+x) + r)/(1 + r*x*(1+x))^2 + x^2*((1+x)^2 + r)^2/(1 + r*x*(1+x)^2)^3 + x^3*((1+x)^3 + r)^3/(1 + r*x*(1+x)^3)^4 + x^4*((1+x)^4 + r)^4/(1 + r*x*(1+x)^4)^5 + x^5*((1+x)^5 + r)^5/(1 + r*x*(1+x)^5)^6 + ...
also,
A(x) = 1/(1 - r*x) + x*((1+x) - r)/(1 - r*x*(1+x))^2 + x^2*((1+x)^2 - r)^2/(1 - r*x*(1+x)^2)^3 + x^3*((1+x)^3 - r)^3/(1 - r*x*(1+x)^3)^4 + x^4*((1+x)^4 - r)^4/(1 - r*x*(1+x)^4)^5 + x^5*((1+x)^5 - r)^5/(1 - r*x*(1+x)^5)^6 + ...

Prog

(PARI) {a(n) = my(r = 2*I, A = sum(m=0, n+1, x^m*((1+x +x*O(x^n))^m + r)^m/(1 + r*x*(1+x +x*O(x^n))^m)^(m+1) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(r = 2*I, A = sum(m=0, n+1, x^m*((1+x +x*O(x^n))^m - r)^m/(1 - r*x*(1+x +x*O(x^n))^m)^(m+1) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A323681, A323682, A323683, A323685.

Sequence in context: A210849 A216155 A011919 * A264529 A263979 A042795

Adjacent sequences: A323681 A323682 A323683 * A323685 A323686 A323687

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2019

Status

approved