A323572 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n + i)^n / (A(x) + 1 + i*(1+x)^n)^(n+1), where i^2 = -1.

1, 5, 45, 1142, 47253, 2664573, 187170069, 15598588065, 1497110942013, 162226788530207, 19566798092698042, 2598785222401424468, 376850999493886187699, 59248452153964672923677, 10039900576546291696149404, 1824412367286993070795917580, 353943959915575446954764374094, 73024199735586268826145811783169, 15966496692824534985042866376857576, 3688160965656359052252569464435170928, 897528733209823570848685886402050648933

Offset

0,2

Comments

It is remarkable that the generating function results in a power series in x with only real coefficients.

Links

Table of n, a(n) for n=0..20.

Formula

G.f. A(x) satisfies:

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

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

(3) 1 = Sum_{n>=0} ((1+x)^n + i)^n * (A(x) + 1 - i*(1+x)^n)^(n+1) / ((A(x) + 1)^2 + (1+x)^(2*n))^(n+1).

(4) 1 = Sum_{n>=0} ((1+x)^n - i)^n * (A(x) + 1 + i*(1+x)^n)^(n+1) / ((A(x) + 1)^2 + (1+x)^(2*n))^(n+1).

Example

G.f.: A(x) = 1 + 5*x + 45*x^2 + 1142*x^3 + 47253*x^4 + 2664573*x^5 + 187170069*x^6 + 15598588065*x^7 + 1497110942013*x^8 + 162226788530207*x^9 + ...
such that
1 = 1/(A(x) + 1+i) + ((1+x) + i)/(A(x) + 1 + i*(1+x))^2  +  ((1+x)^2 + i)^2/(A(x) + 1 + i*(1+x)^2)^3  +  ((1+x)^3 + i)^3/(A(x) + 1 + i*(1+x)^3)^4  +  ((1+x)^4 + i)^4/(A(x) + 1 + i*(1+x)^4)^5  +  ((1+x)^5 + i)^5/(A(x) + 1 + i*(1+x)^5)^6 + ...
also,
1 = 1/(A(x) + 1-i) + ((1+x) - i)/(A(x) + 1 - i*(1+x))^2  +  ((1+x)^2 - i)^2/(A(x) + 1 - i*(1+x)^2)^3  +  ((1+x)^3 - i)^3/(A(x) + 1 - i*(1+x)^3)^4  +  ((1+x)^4 - i)^4/(A(x) + 1 - i*(1+x)^4)^5  +  ((1+x)^5 - i)^5/(A(x) + 1 - i*(1+x)^5)^6 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = round( Vec( sum(m=0, #A*20+300, ((1+x+x*O(x^n))^m + I)^m / (Ser(A) + 1 + I*(1+x+x*O(x^n))^m)^(m+1)*1. ) )[#A]) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = round( Vec( sum(m=0, #A*20+300, ((1+x+x*O(x^n))^m - I)^m / (Ser(A) + 1 - I*(1+x+x*O(x^n))^m)^(m+1)*1. ) )[#A]) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A323313.

Sequence in context: A322661 A191962 A326650 * A368491 A318092 A050641

Adjacent sequences: A323569 A323570 A323571 * A323573 A323574 A323575

Keyword

nonn

Author

Paul D. Hanna, Feb 10 2019

Status

approved