A277311
G.f. satisfies: A(x - 5*A(x)^2) = x - 4*A(x)^2.
Original entry on oeis.org
1, 1, 12, 200, 4034, 92752, 2353272, 64579809, 1891598860, 58591554652, 1906271367296, 64816527248936, 2294331974613872, 84290267670946720, 3206227129084419920, 126022120854865417140, 5110001578581607976400, 213458728365617240931360, 9175021814527973211291880, 405366362599820848509766760, 18392202994173383123235536800, 856255190568423353781484124240
Offset: 1
G.f.: A(x) = x + x^2 + 12*x^3 + 200*x^4 + 4034*x^5 + 92752*x^6 + 2353272*x^7 + 64579809*x^8 + 1891598860*x^9 + 58591554652*x^10 +...
such that A(x - 5*A(x)^2) = x - 4*A(x)^2.
A(x)^2 = x^2 + 2*x^3 + 25*x^4 + 424*x^5 + 8612*x^6 + 198372*x^7 + 5028864*x^8 + 137705810*x^9 + 4022209822*x^10 + 124205854376*x^11 + 4028545272136*x^12 + 136566005356212*x^13 + 4820263259998720*x^14 + 176614868022441920*x^15 +...
A(x - 5*A(x)^2) = x - 4*x^2 - 8*x^3 - 100*x^4 - 1696*x^5 - 34448*x^6 - 793488*x^7 - 20115456*x^8 - 550823240*x^9 - 16088839288*x^10 +...
which equals x - 4*A(x)^2.
Series_Reversion(x - 5*A(x)^2) = x + 5*x^2 + 60*x^3 + 1000*x^4 + 20170*x^5 + 463760*x^6 + 11766360*x^7 + 322899045*x^8 + 9457994300*x^9 +...
which equals 5*A(x) - 4*x.
A( 5*A(x) - 4*x ) = x + 6*x^2 + 82*x^3 + 1525*x^4 + 33864*x^5 + 848402*x^6 + 23259832*x^7 + 685028874*x^8 + 21411099560*x^9 + 704295189492*x^10 +24234549363096*x^11 + 868423052983416*x^12 + 32296557071230392*x^13 + 1243216715481216720*x^14 + 49428242214109804120*x^15 +...
which equals sqrt( A(x) -x ).
Cf.
A277300,
A277301,
A277302,
A277303,
A277304,
A277305,
A277306,
A277307,
A277308,
A277309,
A277310.
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{a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-5*F^2) + 4*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
A276366
G.f. A(x) satisfies: A(x - A(x)^3) = x + A(x)^2.
Original entry on oeis.org
1, 1, 3, 12, 57, 300, 1697, 10126, 62991, 405247, 2680901, 18160444, 125562250, 883868590, 6321838520, 45869309028, 337167193262, 2508018933431, 18861358215299, 143293615189089, 1098997404472941, 8504070741463729, 66358269984208701, 521923129718567918, 4136089275165532156, 33013640650845937124
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 57*x^5 + 300*x^6 + 1697*x^7 + 10126*x^8 + 62991*x^9 + 405247*x^10 + 2680901*x^11 + 18160444*x^12 +...
such that A(x - A(x)^3) = x + A(x)^2.
RELATED SERIES.
A(x - A(x)^3) = x + x^2 + 2*x^3 + 7*x^4 + 30*x^5 + 147*x^6 + 786*x^7 + 4480*x^8 + 26814*x^9 + 166865*x^10 + 1072160*x^11 + 7076724*x^12 +...
which equals x + A(x)^2.
-
{a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-F^3) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
A277033
G.f. A(x) satisfies: A(x - A(-x)^2) = x + A(x)^2.
Original entry on oeis.org
1, 2, 4, 18, 76, 420, 2248, 14410, 89676, 642764, 4487896, 35282228, 271094936, 2310824808, 19309255952, 177093587874, 1596354765308, 15664040851996, 151403517122328, 1582290415072396, 16319413287176584, 180949924453071544, 1983128441367699632, 23249895784026465044, 269763155110100504568, 3333619355332522429656
Offset: 1
G.f.: A(x) = x + 2*x^2 + 4*x^3 + 18*x^4 + 76*x^5 + 420*x^6 + 2248*x^7 + 14410*x^8 + 89676*x^9 + 642764*x^10 +...
such that A(x - A(-x)^2) = x + A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 12*x^4 + 52*x^5 + 240*x^6 + 1288*x^7 + 7108*x^8 + 43908*x^9 + 275872*x^10 + 1904280*x^11 + 13301112*x^12 +...
sqrt((A(x) - x)/2) = x + x^2 + 4*x^3 + 15*x^4 + 82*x^5 + 420*x^6 + 2742*x^7 + 16767*x^8 + 123294*x^9 + 856042*x^10 + 6906790*x^11 + 53066832*x^12 +...
Series_Reversion( sqrt((A(x) - x)/2) ) = x - x^2 - 2*x^3 - 14*x^5 - 406*x^7 - 16514*x^9 - 872812*x^11 - 56605438*x^13 - 4346269882*x^15 - 386603411414*x^17 - 39262351744912*x^19 - 4504838187841052*x^21 -...
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{a(n) = my(A=x,R); for(i=1,n, R = subst(A,x,-x + x*O(x^n)); A = subst(x + A^2,x, serreverse(x - R^2))); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
A277034
G.f. A(x) satisfies: A(x - A(x)^2) = x + A(-x)^2.
Original entry on oeis.org
1, 2, 4, 50, 268, 3780, 28872, 438410, 4087180, 65365260, 697738072, 11624944660, 137432369816, 2371412517480, 30441246407440, 542177876315970, 7460629909188796, 136882304192481020, 2001263659780301080, 37777108180867675020, 583057080531893501960, 11314432259935102732856, 183452721005994056356272
Offset: 1
G.f.: A(x) = x + 2*x^2 + 4*x^3 + 50*x^4 + 268*x^5 + 3780*x^6 + 28872*x^7 + 438410*x^8 + 4087180*x^9 + 65365260*x^10 +...
such that A(x - A(x)^2) = x + A(-x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 12*x^4 + 116*x^5 + 752*x^6 + 9032*x^7 + 77508*x^8 + 1049348*x^9 + 10608800*x^10 + 155499800*x^11 + 1763239416*x^12 +...
sqrt((A(x) - x)/2) = x + x^2 + 12*x^3 + 55*x^4 + 818*x^5 + 5740*x^6 + 92534*x^7 + 815391*x^8 + 13765254*x^9 + 141099882*x^10 + 2462940118*x^11 +...
Series_Reversion( sqrt((A(x) - x)/2) ) = x - x^2 - 10*x^3 - 294*x^5 - 24998*x^7 - 3158794*x^9 - 506665836*x^11 - 96305392110*x^13 - 20904881285306*x^15 - 5068120123901550*x^17 - 1352637633479800560*x^19 - 393510296576306819932*x^21 -...
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{a(n) = my(A=x,R); for(i=1,n, R = subst(A,x,-x + x*O(x^n)); A = subst(x + R^2, x, serreverse(x - A^2 + x*O(x^n)))); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))