A383564 G.f. A(x) satisfies 1 = Sum{n=-oo..+oo} (x^n - 2*x*A(x)^n)^n.

1, 2, 11, 63, 313, 953, -2103, -52455, -340989, -684583, 3097817, -42019812, -959576394, 2335125550, 229613772815, 2356225506958, -11899498155741, -527484556204563, -3363552193233237, 67551575645570526, 1470281680830417628, 2115661347716495378, -323942681936663419906, -4680631899371554723607, 23014994879777225773481

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..250

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

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

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

Example

G.f.: A(x) = 1 + 2*x + 11*x^2 + 63*x^3 + 313*x^4 + 953*x^5 - 2103*x^6 - 52455*x^7 - 340989*x^8 - 684583*x^9 + 3097817*x^10 + ...
where
1 = ... + (1/x^3 - 2*x/A(x)^3)^(-3) + (1/x^2 - 2*x/A(x)^2)^(-2) + (1/x - 2*x/A(x))^(-1) + 1 + (x - 2*x*A(x)) + (x^2 - 2*x*A(x)^2)^2 + (x^3 - 2*x*A(x)^3)^3 + (x^4 - 2*x*A(x)^4)^4 + (x^5 - 2*x*A(x)^5)^5 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoef( sum(n=-#A, #A, (x^n - 2*x*Ser(A)^n)^n )/2, #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A383565.

Sequence in context: A365131 A002629 A235937 * A065928 A188648 A114175

Adjacent sequences: A383561 A383562 A383563 * A383565 A383566 A383567

Keyword

sign

Author

Paul D. Hanna, May 21 2025

Status

approved