A324619 G.f. A(x) = Sum_{n>=0} x^n*(A(x)^n + i)^n/(1 + i*x*A(x)^n)^(n+1), where i^2 = -1.

1, 1, 2, 9, 43, 238, 1436, 9230, 62347, 438397, 3188687, 23886061, 183706628, 1447476799, 11666912970, 96101351315, 808489267438, 6944890495515, 60909005722973, 545480514290103, 4989651763986766, 46633629693716933, 445474644772366480, 4351015655544312048, 43463615209598548901, 444132103382081004621, 4642827326458824657758, 49649335888812297089157, 543030381664721843558408, 6072677103962177819391932

Offset

0,3

Comments

Notice that the imaginary components in the generating function vanish when expanded as a power series in x.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies:

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

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

From Paul D. Hanna, Nov 05 2021: (Start)

(3) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} i^k * binomial(n,k) * (A(x)^n - i*A(x)^k)^(n-k).

(4) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} (-i)^k * binomial(n,k) * (A(x)^n + i*A(x)^k)^(n-k). (End)

Example

G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 43*x^4 + 238*x^5 + 1436*x^6 + 9230*x^7 + 62347*x^8 + 438397*x^9 + 3188687*x^10 + 23886061*x^11 + 183706628*x^12 + ...
such that
A(x) = 1/(1 - i*x) + x*(A(x) - i)/(1 - i*x*A(x))^2 + x^2*(A(x)^2 - i)^2/(1 - i*x*A(x)^2)^3 + x^3*(A(x)^3 - i)^3/(1 - i*x*A(x)^3)^4 + x^4*(A(x)^4 - i)^4/(1 - i*x*A(x)^4)^5 + x^5*(A(x)^5 - i)^5/(1 - i*x*A(x)^5)^6 + ...
also
A(x) = 1/(1 + i*x) + x*(A(x) + i)/(1 + i*x*A(x))^2 + x^2*(A(x)^2 + i)^2/(1 + i*x*A(x)^2)^3 + x^3*(A(x)^3 + i)^3/(1 + i*x*A(x)^3)^4 + x^4*(A(x)^4 + i)^4/(1 + i*x*A(x)^4)^5 + x^5*(A(x)^5 + i)^5/(1 + i*x*A(x)^5)^6 + ...

Prog

(PARI) {a(n) = my(A=[1, 1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff( sum(n=0, #A, x^n*(Ser(A)^n + I)^n/(1 + I*x*Ser(A)^n)^(n+1) ), #A-1)); polcoeff(Ser(A), n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A324618.

Sequence in context: A006795 A055824 A334680 * A292099 A020113 A345414

Adjacent sequences: A324616 A324617 A324618 * A324620 A324621 A324622

Keyword

nonn

Author

Paul D. Hanna, Mar 12 2019

Status

approved