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A323683 G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(3)*i)^n / (1 + sqrt(3)*i*x*(1+x)^n)^(n+1), where i^2 = -1. 5
1, 1, 2, 11, 34, 147, 741, 3723, 20622, 122611, 765147, 5039624, 34856671, 251799740, 1895620770, 14834210103, 120368044792, 1010721096231, 8766998042793, 78426199305392, 722487492991540, 6845217410165959, 66620876210526469, 665308346568565094, 6810637828466836635, 71400562836982319210, 765934679671944152100 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Note that the generating function expands into a power series in x with only real integer 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 + sqrt(3)*i)^n / (1 + sqrt(3)*i*x*(1+x)^n)^(n+1).

G.f.: Sum_{n>=0} x^n * ((1+x)^n - sqrt(3)*i)^n / (1 - sqrt(3)*i*x*(1+x)^n)^(n+1).

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 34*x^4 + 147*x^5 + 741*x^6 + 3723*x^7 + 20622*x^8 + 122611*x^9 + 765147*x^10 + 5039624*x^11 + 34856671*x^12 + ...

Let r = sqrt(3)*i, so that r^2 = -3, 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 = sqrt(3)*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) )); round(polcoeff(A, n))}

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n) = my(r = sqrt(3)*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) )); round(polcoeff(A, n))}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A323681, A323682, A323684, A323685.

Sequence in context: A222650 A026961 A026971 * A071801 A026981 A027223

Adjacent sequences:  A323680 A323681 A323682 * A323684 A323685 A323686

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 12 2019

STATUS

approved

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Last modified June 18 10:54 EDT 2021. Contains 345098 sequences. (Running on oeis4.)