A380557 G.f. satisfies A(x) such that: -1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.

1, 1, 2, 10, 35, 146, 589, 2521, 10880, 48130, 215490, 978131, 4483493, 20740309, 96667511, 453596099, 2140879339, 10157274086, 48414142443, 231726319442, 1113290775079, 5366873616498, 25952658569610, 125856499026093, 611930422986515, 2982444057333882, 14568259180879990, 71307949455547118

Offset: 0

Author

Paul D. Hanna, Feb 03 2025

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1).

Note that formulas (5) and (6) are identities, and hold true for all A(x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 35*x^4 + 146*x^5 + 589*x^6 + 2521*x^7 + 10880*x^8 + 48130*x^9 + 215490*x^10 + ...
SPECIFIC VALUES.
A(t) = 15/8 at t = 0.19344501240894726710748422307503613491843235983978...
  where -1 = Sum_{n=-oo..+oo} (-1)^n * t^(2*n+1) * (1 + t^n)^(n+1) * (15/8)^n.
A(t) = 13/7 at t = 0.19341948378562934846535490742010025888491204467175...
A(t) = 11/6 at t = 0.19333502076470314454576717568898264806286912451280...
A(t) = 9/5 at t = 0.193110973645115287451084966528093291445869685605026...
A(t) = 7/4 at t = 0.192511645242345015361112270688385360547743653185979...
A(t) = 5/3 at t = 0.190649553303712199475798636706794101316873079578727...
A(t) = 3/2 at t = 0.182089586086018008207410926078691444238166561231377...
A(t) = 4/3 at t = 0.161675866655112310035152981730415472582224089685922...
A(t) = 5/4 at t = 0.143001255997678107192529149738503806026990657450325...
A(t) = 6/5 at t = 0.127286533527611786785145642678412658294861536180040...
A(t) = 7/6 at t = 0.114247661034580905508079370420172649525885310579285...
A(1/6) = 1.3631240552377275579566206545056633589020532732074...
  where -1 = Sum_{n=-oo..+oo} (-1)^n * (1/6)^(2*n+1) * (1 + (1/6)^n)^(n+1) * A(1/6)^n.
A(1/7) = 1.2494768685846922246570903862376666502561254090745...
A(1/8) = 1.1937090558071312140144246419584402049019509862828...
A(1/9) = 1.1594652229281839152092617957390758203214375656645...
A(1/10) = 1.135997746902180909378823046338236460433675615420...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A356783.

Programs

PARI

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

Formula

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

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

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

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

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

(5) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n*A(x))^(n+1) / A(x)^n.

(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (A(x) + x^n)^(n-1).