A378260 - OEIS

%I #12 Nov 30 2024 10:28:11

%S 1,1,3,4,11,15,33,50,104,161,309,500,929,1529,2757,4620,8207,13874,

%T 24353,41478,72327,123687,214685,368232,637430,1095201,1892492,

%U 3255372,5619323,9672701,16685587,28734098,49547095,85347087,147130261,253480414,436911525,752798677,1297444411,2235633198

%N G.f. satisfies A(x) = A(x^2)/M(x), where M(x) = Sum_{n>=1} mu(n)*x^n and mu(n) = A008683(n), the Moebius function of n.

%H Paul D. Hanna, <a href="/A378260/b378260.txt">Table of n, a(n) for n = 1..3000</a>

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

%F (1) A(x) = A(x^2)/M(x), where M(x) = Sum_{n>=1} mu(n)*x^n.

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

%F a(n) ~ c * d^n, where d = 1.723262561763844024160437963573163520188527015264827413326383054228438457576... and c = 0.7859046910881843332272010625259660209978142303560254864659049088867251443... - _Vaclav Kotesovec_, Nov 30 2024

%e G.f. A(x) = x + x^2 + 3*x^3 + 4*x^4 + 11*x^5 + 15*x^6 + 33*x^7 + 50*x^8 + 104*x^9 + 161*x^10 + 309*x^11 + 500*x^12 + 929*x^13 + 1529*x^14 + 2757*x^15 + 4620*x^16 + ...

%e where A(x) = A(x^2)/M(x) with

%e M(x) = x - x^2 - x^3 - x^5 + x^6 - x^7 + x^10 - x^11 - x^13 + x^14 + x^15 - x^17 - x^19 + x^21 + x^22 - x^23 + x^26 - x^29 - x^30 + ... + mu(n)*x^n + ...

%e so that x = M(x) + M(x^2) + M(x^3) + M(x^4) + ... + M(x^n) + ...

%e Thus, because M(x) = A(x^2)/A(x), we have

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

%e SPECIFIC VALUES.

%e A(t) = 1000 at t = 0.57983979082390078033201288097053684588681918658...

%e A(t) = 100 at t = 0.575850800621842491687274688724496083876096493693...

%e A(t) = 10 at t = 0.5429296775693301210019293351373468274776922745760...

%e A(t) = 9 at t = 0.53946231343810887800940222774498269502147986174360...

%e A(t) = 8 at t = 0.53525852440539581430297764508815311813586247192451...

%e A(t) = 7 at t = 0.53004645173922704662750351997680689150327151199058...

%e A(t) = 6 at t = 0.52339661111093477495939037490084005628700411644551...

%e A(t) = 5 at t = 0.51458419720941955692565375903201066787604036604586...

%e A(t) = 4 at t = 0.50227142127888616541434068019839636042944372636880...

%e A(t) = 3 at t = 0.48364898724179834772275350279540495722010623952244...

%e A(t) = 2 at t = 0.45148154417138074188660255689175385165406842883889...

%e A(t) = 1 at t = 0.37847838037693933849966786108068785599206753365459...

%e A(1/2) = 3.85113240762543882840278502418639089248043784485031...

%e   where A(1/2) = A(1/4)/M(1/2)

%e   with M(1/2) = 0.10201133481781036474303639393182435154361049251029...

%e A(1/3) = 0.70553754549458547877689262864744328280095059724850...

%e   where A(1/3) = A(1/9)/M(1/3)

%e   with M(1/3) = 0.18199538670263388782780010030056557322634498013538...

%e A(1/4) = 0.39285915746199878617465323026428187937371048080708...

%e   where A(1/4) = A(1/16)/M(1/4)

%e   with M(1/4) = 0.17108224791836356794497287128799432329181231331328...

%e A(1/5) = 0.27550965922396685715103103981428480321441405929553...

%e   where A(1/5) = A(1/25)/M(1/5)

%e   with M(1/5) = 0.15173128129604728456076208173747135942418710339130...

%e A(1/6) = 0.13414853338170816574291660065981488877610508998414...

%e A(1/9) = 0.12840457842551423371933936516424287719901492174905...

%e A(1/16) = 0.06721122777391310699668932733909384687045264984777...

%e A(1/25) = 0.04180343360348984362058625595257513524070610827394...

%o (PARI) {a(n) = my(A=x, M = sum(m=1,n,moebius(m)*x^m) +x*O(x^n));

%o for(i=1,#binary(n), A = subst(A,x,x^2)/M ); polcoef(A,n)}

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

%Y Cf. A073776, A008683.

%K nonn,new

%O 1,3

%A _Paul D. Hanna_, Nov 25 2024