A359712 a(n) = coefficient of x^n in A(x) such that 2 = Sum_{n=-oo..+oo} (-x)^n * (2*A(x) + x^(n-1))^(n+1).
1, 4, 20, 106, 586, 3356, 19728, 118382, 722208, 4466050, 27931600, 176371300, 1122867012, 7199842666, 46454345844, 301384205640, 1964899532794, 12866563846920, 84585757496444, 558060746899684, 3693810227983576, 24521903234307786, 163234951757526400
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 4*x + 20*x^2 + 106*x^3 + 586*x^4 + 3356*x^5 + 19728*x^6 + 118382*x^7 + 722208*x^8 + 4466050*x^9 + 27931600*x^10 + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..200
Programs
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PARI
{a(n) = my(A=1,y=2); for(i=1,n, A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) ); polcoeff( A,n,x)} for(n=0,25, print1( a(n),", ")) -
PARI
{a(n) = my(A=[1],y=2); for(i=1,n, A = concat(A,0); A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]} for(n=0,25, print1( a(n),", "))
Formula
G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(n-1))^(n+1).
(2) 2*x = Sum_{n=-oo..+oo} (-1)^n * (2*x*A(x) + x^n)^(n+1).
(3) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^(n-1).
(4) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * (2*x*A(x) + x^n)^n ].
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^(n+1))^n ].
From Paul D. Hanna, May 12 2023: (Start)
(6) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (2*A(x) + x^n)^n.
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (2*A(x) + x^n)^n ].
(8) 2*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^n. (End)
a(n) = Sum_{k=0..n} A359670(n,k)*2^k for n >= 0.