OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
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.
EXAMPLE
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 + ...
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 17 2023
STATUS
approved