A377249 - OEIS

%I #13 Nov 14 2024 11:46:03

%S 1,3,5,11,20,38,67,119,211,398,830,1940,4902,12784,33165,84136,207240,

%T 495964,1157767,2654461,6029627,13704225,31463620,73498385,175220708,

%U 425631952,1048102141,2599306042,6453178098,15967452038,39281184601,96019973309,233425343306,565413231173

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

%C a(n+1)/a(n) tends to 2.55118... - _Vaclav Kotesovec_, Nov 14 2024

%H Paul D. Hanna, <a href="/A377249/b377249.txt">Table of n, a(n) for n = 1..502</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} (x^(2*n) - A(x))^n.

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

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

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

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

%e G.f.: A(x) = x + 3*x^2 + 5*x^3 + 11*x^4 + 20*x^5 + 38*x^6 + 67*x^7 + 119*x^8 + 211*x^9 + 398*x^10 + 830*x^11 + 1940*x^12 + ...

%e RELATED SERIES.

%e Let A = A(x), then 1 - x = P + Q where

%e P = 1 + (x^2 - A) + (x^4 - A)^2 + (x^6 - A)^3 + (x^8 - A)^4 + (x^10 - A)^5 + (x^12 - A)^6 + ... + (x^(2*n) - A)^n + ...

%e Q = x^2/(1 - x^2*A) + x^8/(1 - x^4*A)^2 + x^18/(1 - x^6*A)^3 + x^32/(1 - x^8*A)^4 + x^50/(1 - x^10*A)^5 + ... + x^(2*n^2)/(1 - x^(2*n)*A)^n + ...

%e Explicitly,

%e P = 1 - x - x^2 - x^5 - 3*x^6 - 5*x^7 - 13*x^8 - 26*x^9 - 57*x^10 - 120*x^11 - 259*x^12 - 561*x^13 - 1238*x^14 - 2780*x^15 + ...

%e Q = x^2 + x^5 + 3*x^6 + 5*x^7 + 13*x^8 + 26*x^9 + 57*x^10 + 120*x^11 + 259*x^12 + 561*x^13 + 1238*x^14 + 2780*x^15 + ...

%e SPECIFIC VALUES.

%e A(t) = 1 at t = 0.31160833954190659544044203165981407225865730702613...

%e   notice that Sum_{n=-oo..+oo} (t^(2*n) - 1)^n = 3/2 - t, which deviates from 1 - t; this is due to the term (t^(2*n) - 1)^n taking on the indeterminate form 0^0 at n = 0.

%e A(t) = 3/4 at t = 0.27771666015004017042369933635782020387857285503045...

%e   where 1-t = Sum_{n=-oo..+oo} (t^(2*n) - 3/4)^n.

%e A(t) = 2/3 at t = 0.26361225583793268826267306259869479301133547781221...

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

%e A(t) = 1/2 at t = 0.22935306806508598431890965331979164445097661571539...

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

%e A(t) = 1/3 at t = 0.18321842757370009004270801439324522647367156844515...

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

%e A(t) = 1/4 at t = 0.15319590401223075722696025027321147967336641684269...

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

%e A(1/4) = 0.59500579234891734482663421150554161897595924474890...

%e   where 3/4 = Sum_{n=-oo..+oo} (1/4^(2*n) - A(1/4))^n.

%e A(1/5) = 0.38777623383308495901886663021576078479818829432566...

%e   where 4/5 = Sum_{n=-oo..+oo} (1/5^(2*n) - A(1/5))^n.

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

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

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

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 14 2024