A107092
G.f. A(x) satisfies A(x)^3 = A(x^3) + 3*x.
Original entry on oeis.org
1, 1, -1, 2, -4, 9, -22, 55, -142, 376, -1011, 2758, -7614, 21220, -59630, 168759, -480533, 1375676, -3957075, 11430582, -33144264, 96434321, -281447954, 823734157, -2417092933, 7109265120, -20955593252, 61893804180, -183148075432, 542885589115, -1611809502764, 4792612539375
Offset: 0
A(x)^3 = 1 + 3*x + x^3 - x^6 + 2*x^9 - 4*x^12 + 9*x^15 - 22*x^18 +...
A(x^3) = 1 + x^3 - x^6 + 2*x^9 - 4*x^12 + 9*x^15 - 22*x^18+...
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{a(n)=local(A=1+x);for(i=1,n,A=(subst(A,x,x^3)+3*x+x*O(x^n))^(1/3)); polcoeff(A,n,x)}
A264228
G.f. A(x) satisfies: A(x)^3 = A( x^3/(1-3*x) ), with A(0) = 0.
Original entry on oeis.org
1, 1, 2, 5, 13, 35, 97, 274, 785, 2275, 6656, 19630, 58295, 174175, 523238, 1579584, 4789919, 14584723, 44577799, 136732988, 420784888, 1298937282, 4021383654, 12483820395, 38853994422, 121220646116, 379062880051, 1187912517953, 3730305167438, 11736596024002, 36994041916973, 116807229667919, 369415244627269, 1170113816365089
Offset: 1
G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 + 58295*x^13 + 174175*x^14 + ...
where A(x)^3 = A( x^3/(1-3*x) ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 87*x^7 + 270*x^8 + 839*x^9 + 2610*x^10 + 8127*x^11 + 25331*x^12 + 79035*x^13 + 246852*x^14 + 771808*x^15 + ...
A( x/(1+x+x^2) ) = x + x^4 + 2*x^7 + 6*x^10 + 22*x^13 + 88*x^16 + 367*x^19 + 1570*x^22 + 6843*x^25 + 30271*x^28 + 135530*x^31 + 612852*x^34 + 2794412*x^37 + 12832472*x^40 + ...
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + x + x^2 + x^3 - x^5 - x^6 + 2*x^8 + 3*x^9 - 6*x^11 - 9*x^12 + 20*x^14 + 30*x^15 - 71*x^17 - 110*x^18 + 267*x^20 + 419*x^21 - 1041*x^23 + ...
Let C0(x) and C2(x) be series trisections of B(x), B(x) = C0(x) + x + C2(x):
C0(x) = 1 + x^3 - x^6 + 3*x^9 - 9*x^12 + 30*x^15 - 110*x^18 + 419*x^21 - 1648*x^24 + 6652*x^27 - 27369*x^30 + 114384*x^33 - 484276*x^36 + ...
C2(x) = x^2 - x^5 + 2*x^8 - 6*x^11 + 20*x^14 - 71*x^17 + 267*x^20 - 1041*x^23 + 4168*x^26 - 17047*x^29 + 70902*x^32 + ... + (-1)^(n-1)*A370446(n)*x^(3*n-1) + ...
then C0(x) = x^2/C2(x).
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{a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3/(1-3*x +x*O(x^n))) )^(1/3) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
A361762
Expansion of g.f. A(x) satisfying A(x)^3 = A( x^3/(1 - 3*x)^3 ) / (1 - 3*x).
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 197, 779, 3135, 12709, 51757, 211761, 871022, 3603282, 14992067, 62719588, 263724900, 1114107925, 4726879206, 20135644606, 86099626270, 369492052236, 1591170063412, 6875211016868, 29803706856996, 129607445296468, 565362988510604, 2473576310166981
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 197*x^6 + 779*x^7 + 3135*x^8 + 12709*x^9 + 51757*x^10 + 211761*x^11 + 871022*x^12 + ...
such that A(x)^3 = A( x^3/(1 - 3*x)^3 ) / (1 - 3*x).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 93*x^4 + 333*x^5 + 1271*x^6 + 5064*x^7 + 20673*x^8 + 85460*x^9 + ... + A361763(n+1)*x^n + ...
A( x^3/(1 - 3*x)^3 ) = 1 + x^3 + 9*x^4 + 54*x^5 + 272*x^6 + 1251*x^7 + 5481*x^8 + 23441*x^9 + 99279*x^10 + ...
SPECIFIC VALUES.
A(1/5) = ( 5/2 * A(1/8) )^(1/3) = 1.431256341682946446458148822310720...
A(1/5) = (1 - 3/5)^(-1/3) * (1 - 3/8)^(-1/9) * (1 - 3/125)^(-1/27) * (1 - 3/1815848)^(-1/81) * ...
A(1/6) = ( 2 * A(1/27) )^(1/3) = 1.2765282682430983587479124671832773...
A(1/6) = (1 - 3/6)^(-1/3) * (1 - 3/27)^(-1/9) * (1 - 3/13824)^(-1/27) * (1 - 3/2640087986661)^(-1/81) * ...
A(1/9) = ( 3/2 * A(1/216) )^(1/3) = 1.146494555403917024085906029391966218...
A(1/12) = ( 4/3 * A(1/729) )^(1/3) = 1.101146836396635655557234214350215617...
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{a(n) = my(A=1); for(i=1, #binary(n+1), A = ( subst(A, x, x^3/(1 - 3*x +x*O(x^n))^3 )/(1 - 3*x +x*O(x^n)) )^(1/3) ); polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))
A361765
Expansion of g.f. A(x) satisfying A(x)^5 = A( x^5/(1 - 5*x)^5 ).
Original entry on oeis.org
1, 5, 25, 125, 625, 3126, 15655, 78650, 397625, 2031875, 10553128, 56047040, 306020575, 1723544750, 10015548750, 59871903136, 366244516505, 2278239803025, 14324961668875, 90586470006875, 573925269278169, 3633524853973370, 22949197586894725, 144473478898021750
Offset: 1
G.f.: A(x) = x + 5*x^2 + 25*x^3 + 125*x^4 + 625*x^5 + 3126*x^6 + 15655*x^7 + 78650*x^8 + 397625*x^9 + 2031875*x^10 + 10553128*x^11 + ...
where
A( x^5/(1 - 5*x)^5 ) = x^5 + 25*x^6 + 375*x^7 + 4375*x^8 + 43750*x^9 + 393755*x^10 + 3281500*x^11 + 25788125*x^12 + 193496875*x^13 + ...
which equals A(x)^5.
RELATED SERIES.
Notice that the following fifth root is an integer series
( A(x)/x )^(1/5) = 1 + x + 3*x^2 + 11*x^3 + 44*x^4 + 185*x^5 + 806*x^6 + 3627*x^7 + 16926*x^8 + 82615*x^9 + 425633*x^10 + ... + A361764(n)*x^n + ...
SPECIFIC VALUES.
A(1/7) = A(1/32)^(1/5) = 0.5172818651818402813815396980...
A(1/7) = (1/7) * (1 - 5/7)^(-1) * (1 - 5/32)^(-1/5) * (1 - 5/14348907)^(-1/25) * (1 - 5/14348902^5)^(-1/125) * ...
A(1/8) = A(1/243)^(1/5) = 0.334722270350398633572525135166...
A(1/8) = (1/8) * (1 - 5/8)^(-1) * (1 - 5/243)^(-1/5) * (1 - 5/763633171168)^(-1/25) * (1 - 5/763633171163^5)^(-1/125) * ...
A(1/10) = A(1/3125)^(1/5) = 0.2000640615121819990127352003599...
A(1/10) = (1/10) * (1 - 5/10)^(-1) * (1 - 5/3125)^(-1/5) * (1 - 5/295646655283200000)^(-1/25) * (1 - 5/295646655283199995^5)^(-1/125) * ...
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{a(n) = my(A=x); for(i=1, #binary(n+1), A = ( subst(A, x, x^5/(1 - 5*x +x*O(x^n))^5 ) )^(1/5) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
Showing 1-4 of 4 results.
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