A378830 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^(2*n+1))^(n+1).

1, 1, 4, 11, 42, 153, 610, 2459, 10252, 43409, 187057, 816018, 3600270, 16030706, 71959064, 325263050, 1479270802, 6764056992, 31078467690, 143411844050, 664352697921, 3088451702294, 14403683874057, 67371591912847, 315969606711112, 1485546933158293, 7000321009422438, 33057274408834760

Offset: 1

Author

Paul D. Hanna, Dec 08 2024

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 11*x^4 + 42*x^5 + 153*x^6 + 610*x^7 + 2459*x^8 + 10252*x^9 + 43409*x^10 + 187057*x^11 + 816018*x^12 + ...
RELATED SERIES.
P(x) = Sum_{n>=0} A(x)^n * (1 + A(x)^(2*n+1))^(n+1) = 1 + 2*x + 3*x^2 + 11*x^3 + 37*x^4 + 142*x^5 + 558*x^6 + 2279*x^7 + 9512*x^8 + 40490*x^9 + 174938*x^10 + 765476*x^11 + 3384754*x^12 + ...
Q(x) = Sum_{n>=0} A(x)^(2*n^2) / (1 + A(x)^(2*n+1))^n = 1 + x^2 + 2*x^3 + 9*x^4 + 29*x^5 + 117*x^6 + 448*x^7 + 1840*x^8 + 7587*x^9 + 32176*x^10 + 138061*x^11 + 601438*x^12 + ...
where 1/x = P(x) + Q(x)/A(x).
SPECIFIC VALUES.
The radius of convergence of A(x) is r = 0.20038502229069403084620325...
  where A(r) = 0.4235821265247070959910742044905... is a relative maximum.
A(t) = 1/3 at t = 0.19298266815962402196002999560324674579680552969760...
  where t = 1/Sum_{n=-oo..+oo} 3*(1/3 + 1/3^(2*n))^n.
A(t) = 1/4 at t = 0.17130301690449912394460704626585777982140255712239...
  where t = 1/Sum_{n=-oo..+oo} 4*(1/4 + 1/4^(2*n))^n.
A(t) = 1/5 at t = 0.15033802491039176999109756769243883091099790716207...
  where t = 1/Sum_{n=-oo..+oo} 5*(1/5 + 1/5^(2*n))^n.
A(1/5) = 0.40253057450577597376072876549180046708653413735166...
A(1/6) = 0.23753161877865991223985753270425915308397301411583...
A(1/7) = 0.18514087266912257977795325275140987102038951384383...
A(1/8) = 0.15359127433897786786958778886350143695575182881679...
A(1/10) = 0.1157765216505016369845861113674765481114936795655...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A378264.

Programs

PARI

{a(n) = my(V=[0,1],A); for(i=1,n, V=concat(V,0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A,#A, A^m*(1 + A^(2*m+1))^(m+1) ), #V-3); ); polcoef(A,n)}
for(n=1,40,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

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

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

(3) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^n * (1 + A(x)^(2*n+1))^n.

(4) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(2*n*(n+1)) / (1 + A(x)^(2*n+1))^(n+1).