A378830 - OEIS

%I #10 Dec 09 2024 05:25:22

%S 1,1,4,11,42,153,610,2459,10252,43409,187057,816018,3600270,16030706,

%T 71959064,325263050,1479270802,6764056992,31078467690,143411844050,

%U 664352697921,3088451702294,14403683874057,67371591912847,315969606711112,1485546933158293,7000321009422438,33057274408834760

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

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

%H Paul D. Hanna, <a href="/A378830/b378830.txt">Table of n, a(n) for n = 1..350</a>

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

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

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

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

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

%e 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 + ...

%e RELATED SERIES.

%e 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 + ...

%e 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 + ...

%e where 1/x = P(x) + Q(x)/A(x).

%e SPECIFIC VALUES.

%e The radius of convergence of A(x) is r = 0.20038502229069403084620325...

%e   where A(r) = 0.4235821265247070959910742044905... is a relative maximum.

%e A(t) = 1/3 at t = 0.19298266815962402196002999560324674579680552969760...

%e   where t = 1/Sum_{n=-oo..+oo} 3*(1/3 + 1/3^(2*n))^n.

%e A(t) = 1/4 at t = 0.17130301690449912394460704626585777982140255712239...

%e   where t = 1/Sum_{n=-oo..+oo} 4*(1/4 + 1/4^(2*n))^n.

%e A(t) = 1/5 at t = 0.15033802491039176999109756769243883091099790716207...

%e   where t = 1/Sum_{n=-oo..+oo} 5*(1/5 + 1/5^(2*n))^n.

%e A(1/5) = 0.40253057450577597376072876549180046708653413735166...

%e A(1/6) = 0.23753161877865991223985753270425915308397301411583...

%e A(1/7) = 0.18514087266912257977795325275140987102038951384383...

%e A(1/8) = 0.15359127433897786786958778886350143695575182881679...

%e A(1/10) = 0.1157765216505016369845861113674765481114936795655...

%o (PARI) {a(n) = my(V=[0,1],A); for(i=1,n, V=concat(V,0); A = Ser(V);

%o V[#V] = polcoef( sum(m=-#A,#A, A^m*(1 + A^(2*m+1))^(m+1) ), #V-3); ); polcoef(A,n)}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A378264.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Dec 08 2024