A380712 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(2*n) * (A(x) + x^n)^(n-1).

1, 8, 84, 1040, 14220, 207416, 3163352, 49838112, 804826128, 13251624272, 221630530572, 3754763811696, 64301286803888, 1111314020855608, 19358763742909840, 339542985410593024, 5991328544544083368, 106282296849129147080, 1894330721630908390908, 33907409814314990430864

Offset: 0

Author

Paul D. Hanna, Feb 21 2025

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 8*x + 84*x^2 + 1040*x^3 + 14220*x^4 + 207416*x^5 + 3163352*x^6 + 49838112*x^7 + 804826128*x^8 + 13251624272*x^9 + ...
where 1/2 = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(2*n) * (A(x) + x^n)^(n-1).
SPECIFIC VALUES.
A(t) = 11/4 at t = 0.0516760605367732994895781933476882835083123366991550...
A(t) = 5/2  at t = 0.0512706205731010752236248787488564353175797944139758...
A(t) = 9/4  at t = 0.0500962358404660270765237205868976107655318520352114...
A(t) = 2 at t = 0.0477293025632112771125356682128360981725295094219849155...
A(1/20) = 2.23597314331862522198845196504910911754398888561681...
A(1/21) = 1.99138007575616089662630287849407074435849433028712...
A(1/22) = 1.84963311768563536747915607573430165080208182054100...
A(1/23) = 1.75061850078240549448668878592242454463377302943692...
A(1/24) = 1.67559723922061838865975191745342891467177127438988...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A379763, A380068.

Programs

PARI

{a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef(2 + 4*sum(n=-#V, #V, (-1)^n * x^(2*n) * (A + x^n)^(n-1) ), #V-1) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

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

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

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