A265266 G.f. A(x) satisfies: A(x) = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(2*n).

1, 2, 5, 27, 143, 833, 5198, 33607, 223627, 1522249, 10546221, 74119591, 527150783, 3786896705, 27437431852, 200267244944, 1471209231873, 10869315344076, 80707738490984, 601977204069443, 4508156389422426, 33884634730883602, 255532279985062648, 1932864141175160374, 14660843479381675987, 111486308441258038306, 849773662058395948696, 6491244696415245552638, 49685280480631490670702, 381014689125058139363522, 2926949265189880054761750

Offset: 0

Author

Paul D. Hanna, Jan 03 2016

Keywords

• nonn

Comments

Compare to: Sum_{n=-oo..+oo} x^n * (c - x^n)^n = 0 for fixed |c| > 0.

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 27*x^3 + 143*x^4 + 833*x^5 + 5198*x^6 + 33607*x^7 + 223627*x^8 + 1522249*x^9 + 10546221*x^10 + ...
Let A = g.f. A(x) where A(x) = P(x) + N(x) then
P(x) = 1 + x*(A - x)^2 + x^2*(A - x^2)^4 + x^3*(A - x^3)^6 + x^4*(A - x^4)^8 + x^5*(A - x^5)^10 + x^6*(A - x^6)^12 + x^7*(A - x^7)^14 + x^8*(A - x^8)^16 + ...
N(x) = x/(1-x*A)^2 + x^6/(1-x^2*A)^4 + x^15/(1-x^3*A)^6 + x^28/(1-x^4*A)^8 + x^45/(1-x^5*A)^10 + x^66/(1-x^6*A)^12 + x^91/(1-x^7*A)^14 + ...
Explicitly,
P(x) = 1 + x + 3*x^2 + 20*x^3 + 117*x^4 + 708*x^5 + 4535*x^6 + 29801*x^7 + 200369*x^8 + 1373999*x^9 + 9570641*x^10 + 67539460*x^11 + 481899317*x^12 + ...
N(x) = x + 2*x^2 + 7*x^3 + 26*x^4 + 125*x^5 + 663*x^6 + 3806*x^7 + 23258*x^8 + 148250*x^9 + 975580*x^10 + 6580131*x^11 + 45251466*x^12 + ...
		

Links

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

Crossrefs

Cf. A260147.

Programs

PARI

{a(n) = my(A=[1,1]); for(i=1,n, A=Vec( sum(n=-#A-1,#A+1, x^n*(Ser(A) - x^n)^(2*n) ) ) );A[n+1]}
for(n=0,40,print1(a(n),", "))

PARI

/* Quick print of terms 0..N (informal): */
N = 40; A=[1]; for(i=1,N, A=Vec( sum(n=-#A-1,#A+1, x^n*(Ser(A) - x^n)^(2*n) ) ) );A

Formula

The g.f. A(x) = Sum_{n>=0} a(n)*x^n also satisfies:

(1) A(x) = Sum_{n=-oo..+oo} x^n * (A(x) + x^n)^(2*n).

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

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

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

(5) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-x)^n * (A(x^2) + x^n)^n.

(6) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n*A(x^2))^n.

(7) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n*A(x^2))^n.

a(n) ~ c * d^n / n^(3/2), where d = 8.078575206447883305059904... and c = 0.294232997886629805825... - Vaclav Kotesovec, Sep 03 2017