A372531
Expansion of g.f. A(x) satisfying A(x)^3 = A( x*A(x)^2/(1 - A(x)) ).
Original entry on oeis.org
1, 1, 2, 6, 19, 63, 220, 795, 2942, 11100, 42547, 165204, 648423, 2568522, 10255044, 41226054, 166732446, 677922831, 2769487183, 11362238976, 46794199487, 193387049685, 801742251778, 3333468469185, 13896609698686, 58073938493679, 243238872937589, 1020921149848044
Offset: 1
G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 63*x^6 + 220*x^7 + 795*x^8 + 2942*x^9 + 11100*x^10 + 42547*x^11 + 165204*x^12 + ...
where A( x*A(x)^2/(1 - A(x)) ) = A(x)^3.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 111*x^7 + 405*x^8 + 1511*x^9 + 5742*x^10 + 22131*x^11 + 86310*x^12 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x * Product_{n>=0} (1 - x^(3^n)) = x - x^2 - x^4 + x^5 - x^10 + x^11 + x^13 - x^14 - x^28 + x^29 + x^31 - x^32 + x^37 - x^38 - x^40 + x^41 - x^82 + ... + (-1)^A010060(n-1)*x^(A005836(n) + 1) + ...
thus,
x = A(x) * (1 - A(x)) * (1 - A(x)^3) * (1 - A(x)^9) * (1 - A(x)^27) * (1 - A(x)^81) * ... * (1 - A(x)^(3^n)) * ...
SPECIFIC VALUES.
A(t) = 2/5 at t = (2/5) * Product_{n>=0} (1 - (2/5)^(3^n)) = 0.224581111967794306351236678951339491766788581...
A(t) = 1/3 at t = (1/3) * Product_{n>=0} (1 - 1/3^(3^n)) = 0.213980897639074346024397964153942364246900732...
A(t) = 1/4 at t = (1/4) * Product_{n>=0} (1 - 1/4^(3^n)) = 0.184569608420133580452570741795926229187208705...
A(1/5) = 0.2875735682779125398024437286065851185781152551441974155...
A(1/6) = 0.2142049226274852453913309509157678575015367219972692281...
A(1/7) = 0.1738274438474723142423128822604443845966959546404795042...
A(1/6)^3 = A(t) at t = (1/6)*A(1/6)^2/(1 - A(1/6)) = 0.0097319157371...
A(1/7)^3 = A(t) at t = (1/7)*A(1/7)^2/(1 - A(1/7)) = 0.0052247784954...
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/* From Series_Reversion( x * Product_{n>=0} (1 - x^(3^n)) ) */
{a(n) = my(A, M=ceil(log(n+1)/log(3))); A = serreverse( x * prod(m=0,M, 1 - x^(3^m)) + x*O(x^n) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
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/* From A(x)^3 = A( x*A(x)^2/(1 - A(x)) ) */
{a(n) = my(A=[0, 1],F); for(i=1, n, A = concat(A, 0); F=Ser(A);
A[#A] = polcoeff( subst(F, x, x*F^2/(1 - F) ) - F^3, #A+1) ); H=A; A[n+1]}
for(n=1, 30, print1(a(n), ", "))
A373312
Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x))^2 ).
Original entry on oeis.org
1, 2, 9, 46, 266, 1636, 10529, 69974, 476598, 3309212, 23336626, 166686732, 1203409180, 8767531432, 64378620609, 475951684454, 3539801952222, 26466142669804, 198814291126846, 1499817211781796, 11357495427008900, 86302897747248024, 657858710864911954, 5029067212015246972
Offset: 1
G.f.: A(x) = x + 2*x^2 + 9*x^3 + 46*x^4 + 266*x^5 + 1636*x^6 + 10529*x^7 + 69974*x^8 + 476598*x^9 + 3309212*x^10 + ...
where A( x*A(x)/(1 - A(x))^2 ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 128*x^5 + 797*x^6 + 5164*x^7 + 34506*x^8 + 235984*x^9 + 1643882*x^10 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x * Product_{n>=0} (1 - x^(2^n))^2 = x - 2*x^2 - x^3 + 4*x^4 - 3*x^5 + 2*x^6 + 3*x^7 - 8*x^8 + x^9 + 6*x^10 + ... + A106407(n-1) * x^n + ...
thus,
x = A(x) * (1 - A(x))^2 * (1 - A(x)^2)^2 * (1 - A(x)^4)^2 * (1 - A(x)^8)^2 * (1 - A(x)^16)^2 * ... * (1 - A(x)^(2^n))^2 * ...
Also, notice that the square root of A(x)/x is an integral series
(A(x)/x)^(1/2) = 1 + x + 4*x^2 + 19*x^3 + 106*x^4 + 636*x^5 + 4024*x^6 + 26405*x^7 + 178096*x^8 + 1227018*x^9 + 8598424*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/5 at t = (1/5) * Product_{n>=0} (1 - 1/5^(2^n))^2 = 0.1175870125805304806733576532618445158357121658...
A(t) = 1/6 at t = (1/6) * Product_{n>=0} (1 - 1/6^(2^n))^2 = 0.1092311136132535692899568885022954464596243049...
A(1/9) = 0.17288740832245782814001741323630181133096513764543378...
A(1/10) = 0.1413215396171684711943139566840401836123301177323661...
A(1/11) = 0.1213541857717280074895334383318404648498876032468172...
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{a(n) = my(A = serreverse(x*prod(k=0,#binary(n), (1 - x^(2^k) + x*O(x^n))^2))); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
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{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( subst(Ser(A),x, x*Ser(A)/(1 - Ser(A))^2 ) - Ser(A)^2,#A)); A[n+1]}
for(n=1,30,print1(a(n),", "))
A373313
Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x))^3 ).
Original entry on oeis.org
1, 3, 18, 127, 999, 8376, 73400, 664143, 6157467, 58190531, 558478098, 5428532148, 53331912158, 528721992000, 5282688183600, 53140908294191, 537760961917833, 5470638540940401, 55914705172750446, 573908634206898951, 5913010265931479289, 61132102068652970100, 634002859944973526904
Offset: 1
G.f.: A(x) = x + 3*x^2 + 18*x^3 + 127*x^4 + 999*x^5 + 8376*x^6 + 73400*x^7 + 664143*x^8 + 6157467*x^9 + 58190531*x^10 + ...
where A( x*A(x)/(1 - A(x))^3 ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 45*x^4 + 362*x^5 + 3084*x^6 + 27318*x^7 + 249149*x^8 + 2323968*x^9 + 22067697*x^10 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x * Product_{n>=0} (1 - x^(2^n))^3 = x - 3*x^2 + 8*x^4 - 9*x^5 + 3*x^6 + 8*x^7 - 24*x^8 + 15*x^9 + 19*x^10 + ... + A373308(n-1) * x^n + ...
thus,
x = A(x) * (1 - A(x))^3 * (1 - A(x)^2)^3 * (1 - A(x)^4)^3 * (1 - A(x)^8)^3 * (1 - A(x)^16)^3 * ... * (1 - A(x)^(2^n))^3 * ...
Also, notice that the cube root of A(x)/x is an integral series
(A(x)/x)^(1/3) = 1 + x + 5*x^2 + 32*x^3 + 239*x^4 + 1937*x^5 + 16578*x^6 + 147408*x^7 + 1348465*x^8 + 12608851*x^9 + 119972595*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/6 at t = (1/6) * Product_{n>=0} (1 - 1/6^(2^n))^3 = 0.0884290923082561345726735004152032422138677544...
A(t) = 1/7 at t = (1/7) * Product_{n>=0} (1 - 1/7^(2^n))^3 = 0.0844605844040460521136280418653467784637497846...
A(t) = 1/8 at t = (1/8) * Product_{n>=0} (1 - 1/8^(2^n))^3 = 0.0798174217593180496284155971364088109289815675...
A(1/12) = 0.1379538716718371951653031812720929490038524971492263...
A(1/13) = 0.1160657279647048938238673646663527089747582497393475...
A(1/14) = 0.1016889922856297159061963243507242491941351481713051...
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{a(n) = my(A = serreverse(x*prod(k=0,#binary(n), (1 - x^(2^k) + x*O(x^n))^3))); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
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{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( subst(Ser(A),x, x*Ser(A)/(1 - Ser(A))^3 ) - Ser(A)^2,#A)); A[n+1]}
for(n=1,30,print1(a(n),", "))
A372534
Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x) + 3*x*A(x)^2 ).
Original entry on oeis.org
1, 3, 12, 63, 372, 2322, 15102, 101439, 698340, 4900914, 34931808, 252185238, 1840242546, 13551558336, 100579610790, 751610709279, 5650352546628, 42702935642082, 324256445598816, 2472613511240754, 18926918200655928, 145379893260849876, 1120198916414984148, 8656357557290045382
Offset: 1
G.f.: A(x) = x + 3*x^2 + 12*x^3 + 63*x^4 + 372*x^5 + 2322*x^6 + 15102*x^7 + 101439*x^8 + 698340*x^9 + 4900914*x^10 + 34931808*x^11 + 252185238*x^12 + ...
where A(x)^2 = A( x*A(x) + 3*x*A(x)^2 ).
Also,
A(x) = x * (1 + 3*A(x)) * (1 + 3*A(x)^2) * (1 + 3*A(x)^4) * (1 + 3*A(x)^8) * (1 + 3*A(x)^16) * ... * (1 + 3*A(x)^(2^n)) * ...
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 33*x^4 + 198*x^5 + 1266*x^6 + 8388*x^7 + 57033*x^8 + 396090*x^9 + 2798718*x^10 + 20056824*x^11 + 145438146*x^12 + ...
x*A(x) + 3*x*A(x)^2 = x^2 + 6*x^3 + 30*x^4 + 162*x^5 + 966*x^6 + 6120*x^7 + 40266*x^8 + 272538*x^9 + 1886610*x^10 + 13297068*x^11 + ...
SPECIFIC VALUES.
A(1/9) = 0.20017482594200170883488591841314367600913783...
A(1/10) = 0.15939222988059047986391116283589184626082823...
A(1/11) = 0.13474373940944085584086064879196682498369755...
A(1/12) = 0.11741441277153705906655653078308588616286400...
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{a(n) = my(A = serreverse( x/prod(k=0, #binary(n), 1 + 3*x^(2^k) +x*O(x^n)) ));
polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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{a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0); F=Ser(A);
A[#A] = polcoeff( subst(F, x, x*F*(1 + 3*F) ) - F^2, #A) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
A372532
Expansion of g.f. A(x) satisfying A(x)^4 = A( x*A(x)^3/(1 - A(x)) ).
Original entry on oeis.org
1, 1, 2, 5, 15, 48, 160, 549, 1930, 6919, 25200, 92976, 346757, 1305140, 4951216, 18912245, 72675115, 280761688, 1089800460, 4248151335, 16623220558, 65273436984, 257115848688, 1015721354400, 4023189912040, 15974444935191, 63571105091684, 253513322846012, 1012942417348605
Offset: 1
G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 48*x^6 + 160*x^7 + 549*x^8 + 1930*x^9 + 6919*x^10 + 25200*x^11 + 92976*x^12 + ...
where A(x)^4 = A( x*A(x)^3/(1 - A(x)) ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 93*x^7 + 321*x^8 + 1136*x^9 + 4092*x^10 + 14955*x^11 + 55328*x^12 + 206829*x^13 + ...
A(x)^4 = x^4 + 4*x^5 + 14*x^6 + 48*x^7 + 169*x^8 + 608*x^9 + 2222*x^10 + 8216*x^11 + 30680*x^12 + 115556*x^13 + 438554*x^14 + ...
x*A(x)^3/(1 - A(x)) = x^4 + 4*x^5 + 14*x^6 + 48*x^7 + 168*x^8 + 600*x^9 + 2178*x^10 + 8008*x^11 + 29762*x^12 + 111644*x^13 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x * Product_{n>=0} (1 - x^(4^n)) = x - x^2 - x^5 + x^6 - x^17 + x^18 + x^21 - x^22 - x^65 + x^66 + x^69 - x^70 + x^81 - x^82 - x^85 + x^86 - x^257 + x^258 + x^261 + ... + (-1)^A010060(n)*x^(A000695(n) + 1) + ...
thus,
x = A(x) * (1 - A(x)) * (1 - A(x)^4) * (1 - A(x)^16) * (1 - A(x)^64) * (1 - A(x)^256) * ... * (1 - A(x)^(4^n)) * ...
SPECIFIC VALUES.
A(t) = 2/5 at t = (2/5) * Product_{n>=0} (1 - (2/5)^(4^n)) = 0.2338558995596128026623999920422960979429704653...
A(t) = 1/3 at t = (1/3) * Product_{n>=0} (1 - 1/3^(4^n)) = 0.2194787328986396432551386254242908520274591882...
A(t) = 1/4 at t = (1/4) * Product_{n>=0} (1 - 1/4^(4^n)) = 0.1867675780815147845714818686246871948236698894...
A(t) = 1/5 at t = (1/5) * Product_{n>=0} (1 - 1/5^(4^n)) = 0.1597439999989531017215999999999999999999999997...
A(1/5) = 0.2791419705799491640241970731636463821918278265598702481...
A(1/6) = 0.2119087418184569371633725749849800368394048924883176302...
A(1/7) = 0.1728682698948146927220680877897385568988140527227279611...
A(1/6)^4 = A(t) at t = (1/6)*A(1/6)^3/(1 - A(1/6)) = 0.0020124210815...
A(1/7)^4 = A(t) at t = (1/7)*A(1/7)^3/(1 - A(1/7)) = 0.0008922224261..
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/* From Series_Reversion( x * Product_{n>=0} (1 - x^(4^n)) ) */
{a(n) = my(A, M=ceil(log(n+1)/log(4))); A = serreverse( x * prod(m=0, M, 1 - x^(4^m)) + x*O(x^n) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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/* From A(x)^4 = A( x*A(x)^3/(1 - A(x)) ) */
{a(n) = my(A=[0, 1], F); for(i=1, n, A = concat(A, 0); F=Ser(A);
A[#A] = polcoeff( subst(F, x, x*F^3/(1 - F) ) - F^4, #A+2) ); H=A; A[n+1]}
for(n=1, 30, print1(a(n), ", "))
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