A383565 G.f. A(x) satisfies 1 = Sum{n=-oo..+oo} (2*x*A(x)^(n-2) - x^n)^n.

1, 2, 5, 51, 420, 4281, 46511, 545096, 6692705, 85823095, 1139652239, 15617770854, 220169697660, 3186889572660, 47301102618948, 719343250099901, 11204504778316013, 178732002565811126, 2920246793283514209, 48883298337473391637, 838635194555095619608, 14750589639992176103092, 266072885796042137133800

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..301

Formula

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

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

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

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 51*x^3 + 420*x^4 + 4281*x^5 + 46511*x^6 + 545096*x^7 + 6692705*x^8 + 85823095*x^9 + 1139652239*x^10 + ...
where
1 = ... + (2*x/A(x)^4 - 1/x^2)^(-2) + (2*x/A(x)^3 - 1/x)^(-1) + 1 + (2*x/A(x) - x) + (2*x - x^2)^2 + (2*x*A(x) - x^3)^3 + (2*x*A(x)^2 - x^4)^4 + (2*x*A(x)^3 - x^5)^5 + (2*x*A(x)^4 - x^6)^6 + ...

Prog

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

Crossrefs

Sequence in context: A208797 A004098 A356492 * A208206 A376043 A005114

Adjacent sequences: A383562 A383563 A383564 * A383566 A383567 A383568

Keyword

nonn

Author

Paul D. Hanna, May 21 2025

Status

approved