A380709
G.f. A(x) satisfies A(x) = 1 + x*abs( 1/A(x) )^3.
Original entry on oeis.org
1, 1, 3, 9, 25, 60, 111, 356, 717, 1728, 3532, 7923, 13947, 43956, 135762, 455844, 1502005, 4377084, 9696816, 33777040, 76261380, 211981800, 491690441, 1156806114, 2388107247, 7425085120, 22208783472, 72885740508, 243066599038, 726160343256, 1695120635568, 5836780502656, 13416367141485
Offset: 0
G.f.: A(x) 1 + x + 3*x^2 + 9*x^3 + 25*x^4 + 60*x^5 + 111*x^6 + 356*x^7 + 717*x^8 + 1728*x^9 + 3532*x^10 + 7923*x^11 + 13947*x^12 + ...
RELATED SERIES.
1/A(x) = 1 - x - 2*x^2 - 4*x^3 - 6*x^4 + x^5 + 52*x^6 - 26*x^7 + 112*x^8 - 22*x^9 + 280*x^10 + 266*x^11 + 3978*x^12 + ...
The absolute value of the series 1/A(x) begins
abs(1/A(x)) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + x^5 + 52*x^6 + 26*x^7 + 112*x^8 + 22*x^9 + 280*x^10 + 266*x^11 + 3978*x^12 + 19476*x^13 + 53748*x^14 + 188096*x^15 + 356128*x^16 + 145318*x^17 + 4083268*x^18 + ...
the cube of which starts as
abs(1/A(x))^3 = 1 + 3*x + 9*x^2 + 25*x^3 + 60*x^4 + 111*x^5 + 356*x^6 + 717*x^7 + 1728*x^8 + ...
where A(x) = 1 + x*abs(1/A(x))^3.
SPECIFIC VALUES.
A(t) = 5 at t = 0.34652481192452632778148744009...
A(t) = 9/2 at t = 0.34332047911369115853530109434629340595421524344707...
A(t) = 4 at t = 0.33844988613244193281810217915341671138001247109315...
A(t) = 7/2 at t = 0.33093206633015553479076876378936259852312274709636...
A(t) = 3 at t = 0.31913094940940804614787566004609274666160372407803...
A(t) = 5/2 at t = 0.30017933266419626029599691715268323619028106096701...
A(t) = 2 at t = 0.26823879592468130644447947201722810537538246719689...
A(t) = 3/2 at t = 0.20641070526053514308343007863179336080812858639439...
A(1/3) = 3.6370099291721444216320225286542434877849899595617...
A(1/4) = 1.8094747379526694743161159394189701882898513040217...
A(1/5) = 1.4662568572713513624196239629654486684279393066965...
A(1/6) = 1.3230157298226165571635234305575666232122775793769...
A(1/7) = 1.2458642715965738773970674152984414596827918944570...
A(1/8) = 1.1980410385476832715212621689007173781378273728475...
-
{a(n) = my(A=1); for(i=1,n, A = 1 + x*Ser(abs(Vec(1/(A +x*O(x^n)))))^3 ); polcoef(H=A,n)}
for(n=0,40,print1(a(n),", "))
A380710
G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ).
Original entry on oeis.org
1, 1, 3, 8, 19, 52, 130, 350, 887, 2386, 6178, 16318, 42618, 112632, 295072, 777628, 2039543, 5379446, 14139050, 37212510, 97869194, 257724328, 677880176, 1784741604, 4694887026, 12362045980, 32529481476, 85628088892, 225332403940, 593217232816, 1561270271280, 4109624293656, 10816272052191
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 19*x^4 + 52*x^5 + 130*x^6 + 350*x^7 + 887*x^8 + 2386*x^9 + 6178*x^10 + 16318*x^11 + 42618*x^12 + ...
where A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 7*x^2 + 22*x^3 + 63*x^4 + 190*x^5 + 542*x^6 + 1576*x^7 + 4447*x^8 + 12702*x^9 + 35694*x^10 + ...
1/A(x) = 1 - x - 2*x^2 - 3*x^3 - 2*x^4 - 6*x^5 - 4*x^6 - 21*x^7 - 2*x^8 - 94*x^9 - 52*x^10 - 270*x^11 - 84*x^12 + ...
The absolute value of the series 1/A(x) begins
abs(1/A(x)) = 1 + x + 2*x^2 + 3*x^3 + 2*x^4 + 6*x^5 + 4*x^6 + 21*x^7 + 2*x^8 + 94*x^9 + 52*x^10 + 270*x^11 + 84*x^12 + ...
where abs(1/A(x)) = 2 - 1/A(x).
The absolute value of the series 1/A(x)^2 starts as
abs( 1/A(x)^2 ) = 1 + 2*x + 3*x^2 + 2*x^3 + 6*x^4 + 4*x^5 + 21*x^6 + 2*x^7 + 94*x^8 + 52*x^9 + 270*x^10 + 84*x^11 + 1420*x^12 + ...
where abs(1/A(x)) = 1 + x*abs( 1/A(x)^2 ).
The occurrence of signs in the expansion of 1/A(x)^2 has no obvious pattern.
SPECIFIC VALUES.
A(t) = 6 at t = 0.3521253776792082580595807444750553302449897977015...
A(t) = 5 at t = 0.3456635940865550514487387712620006990835608970892...
A(t) = 4 at t = 0.3353445841109354623507968372790782182828383144865...
A(t) = 3 at t = 0.3163390965835750115994353781504066116184311812558...
A(t) = 2 at t = 0.2703238890812296559650050596866021785482700845665...
A(t) = 3/2 at t = 0.21007722302555848449805443502768527123106826520...
A(1/3) = 3.8561630489436922241277332770003463055663996660504...
A(1/4) = 1.7804507530929577349684197763505149006496008002510...
A(1/5) = 1.4486680710862436038990844874974598495016144300066...
A(1/6) = 1.3133683293052424032190784618054973634892830723346...
A(1/7) = 1.2401905953633440750393755932922609861657646670157...
A(1/8) = 1.1944676144162474770850469959020275729350893069119...
A(1/9) = 1.1632456733394683634583953215349829285608074021805...
A(1/10) = 1.140597094866485485300620048216088625981556459200...
Let B(x) = abs(1/A(x)^2) then B(x) = (1 - 1/A(x))/x with
B(r) = 1/r = 2.63223139752362698799211074224388216591957118984454...
B(1/3) = 2.2220245975279023880827256557743784168347448472346...
B(1/4) = 1.7533779055380819630180398010000489157697985459653...
B(1/5) = 1.5485537371919237697954310652528503110387458910072...
B(1/6) = 1.4315938140719938763450788025656509230188193709550...
B(1/7) = 1.3557062711403811961362651790083206606967946565600...
B(1/8) = 1.3024555011399714687578982713429488443700165977339...
-
{a(n) = my(A=1); for(i=1, n, A = 1 + x*A*Ser(abs(Vec(1/(A^2 +x*O(x^n))))) ); polcoef(A, n)}
for(n=0, 40, print1(a(n), ", "))
A380711
G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).
Original entry on oeis.org
1, 1, 4, 13, 32, 147, 460, 1436, 5662, 17287, 60644, 209377, 688370, 2391256, 8105590, 27102666, 92744010, 312994179, 1067043874, 3659563265, 12430287670, 42225015449, 143808001426, 487301478188, 1658050374982, 5637187122368, 19153301908756, 65251831433398, 222042679730222, 755372323224172
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + 460*x^6 + 1436*x^7 + 5662*x^8 + 17287*x^9 + 60644*x^10 + 209377*x^11 + 688370*x^12 + ...
where A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 9*x^2 + 34*x^3 + 106*x^4 + 462*x^5 + 1639*x^6 + 5800*x^7 + 22722*x^8 + 78754*x^9 + 289543*x^10 + ...
1/A(x) = 1 - x - 3*x^2 - 6*x^3 - x^4 - 51*x^5 - 84*x^6 - 42*x^7 - 891*x^8 - 627*x^9 - 2373*x^10 + ...
The absolute value of the series 1/A(x) begins
abs(1/A(x)) = 1 + x + 3*x^2 + 6*x^3 + x^4 + 51*x^5 + 84*x^6 + 42*x^7 + 891*x^8 + 627*x^9 + 2373*x^10 + 7848*x^11 + 15624*x^12 + ...
where abs(1/A(x)) = 2 - 1/A(x).
The absolute value of the series 1/A(x)^3 starts as
abs( 1/A(x)^3 ) = 1 + 3*x + 6*x^2 + x^3 + 51*x^4 + 84*x^5 + 42*x^6 + 891*x^7 + 627*x^8 + 2373*x^9 + 7848*x^10 + 15624*x^11 + ...
where abs(1/A(x)) = 1 + x*abs( 1/A(x)^3 ).
The occurrence of signs in the expansion of 1/A(x)^3 has no obvious pattern.
SPECIFIC VALUES.
A(t) = 6 at t = 0.2776546403334668208899822312116577117579321589899...
A(t) = 5 at t = 0.27378228956266390389083139456755472304789559095846856286...
A(t) = 4 at t = 0.26751987468975853019031596683845328283581047906415763868...
A(t) = 3 at t = 0.25570653476578627566868647080655632304757429284241743094...
A(t) = 2 at t = 0.22541634177918528190705637551445570310188162066848813268...
A(t) = 3/2 at t = 0.181930310644869474243648515956090159019218115295765171...
A(1/4) = 2.7078534198843535187257007342533795310245294411311514375...
A(1/5) = 1.6527957689077139045813143038292189120779186108157811947...
A(1/6) = 1.4039503414912111190464124769746901176157597824012670753...
A(1/7) = 1.2919470482512907310654123055517832107265014355362879392...
A(1/8) = 1.2281933933225341024142993760196501004863261649342668152...
A(1/9) = 1.1870632020801295908616256565906659737605022656656501307...
A(1/10) = 1.158356714849802903775203606108124940003741201462273033...
Let B(x) = abs(1/A(x)^3) then B(x) = (1 - 1/A(x))/x with
B(r) = 1/r = 3.4018842764560748576093421240750088532575559256068992507649...
B(1/4) = 2.5228151676796334019994272154634466465512689587018470209...
B(1/5) = 1.9748228461981367841496533002109555899836788562358152424...
B(1/6) = 1.7263445702594598152254927641049269014234569252848538480...
B(1/7) = 1.5818212832524217283358477460759387507949300014779006395...
B(1/8) = 1.4863678281494130346658952345106428057380981383649536590...
-
{a(n) = my(A=1); for(i=1, n, A = 1 + x*A*Ser(abs(Vec(1/(A^3 +x*O(x^n))))) ); polcoef(H=A, n)}
for(n=0, 40, print1(a(n), ", "))
A384267
G.f. A(x) satisfies A(x) = 1 + abs( x/A(x)^2 ).
Original entry on oeis.org
1, 1, 2, 1, 6, 13, 4, 80, 242, 109, 1702, 5177, 2208, 40348, 128560, 56864, 1052102, 3406333, 1509862, 28900645, 94971462, 42420281, 825816148, 2740269448, 1228678588, 24277298940, 81183221736, 36526643608, 729682028652, 2454721201940, 1107304048024, 22319301025880, 75450489469554
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 6*x^4 + 13*x^5 + 4*x^6 + 80*x^7 + 242*x^8 + 109*x^9 + 1702*x^10 + 5177*x^11 + 2208*x^12 + ...
RELATED SERIES.
A(x) equals the series formed from the absolute values of the coefficients in 1 + x/A(x)^2 where
x/A(x)^2 = x - 2*x^2 - x^3 + 6*x^4 - 13*x^5 - 4*x^6 + 80*x^7 - 242*x^8 - 109*x^9 + 1702*x^10 - 5177*x^11 - 2208*x^12 +-- ...
notice that the signs in x/A(x)^2 seem to be {+,-,-} repeating.
SPECIFIC VALUES.
A(t) = 17/10 at t = 0.2988099109194334966744754680560903...
A(t) = 8/5 at t = 0.28632536002959676347841744332637502584281553236328...
A(t) = 3/2 at t = 0.26584952269781748463288503061262604182943155912168...
A(t) = 7/5 at t = 0.23679807527229400928334910529482907166586736528066...
A(t) = 4/3 at t = 0.21222131698512068142939257924460486238379301612052...
A(t) = 6/5 at t = 0.14851037601497632663099987292554419705752970437155...
A(1/4) = 1.4416840609369316144418746432100574811353758654573...
A(1/5) = 1.3040757997934088091953759590684948311334157108446...
A(1/6) = 1.2337286609104904159907289378298492254783023920577...
A(1/7) = 1.1900466603567900992777823995090950832516801703123...
A(1/8) = 1.1601356692672906064760109443886299674930778512606...
A(1/9) = 1.1383397014975021472515053785203604745989973570682...
A(1/10) = 1.1217447587000441822177506555087189442697776256039...
-
{a(n) = my(A=1); for(i=1, n, A = 1 + x*Ser(abs(Vec(1/(A +x*O(x^n))^2))) ); polcoef(A, n)}
for(n=0, 32, print1(a(n), ", "))
Showing 1-4 of 4 results.
Comments