A374565
Expansion of g.f. A(x) satisfying A(x)^3 = A( x*A(x)^2/(1-x) ).
Original entry on oeis.org
1, 1, 1, 2, 5, 11, 24, 57, 141, 350, 881, 2267, 5920, 15601, 41497, 111399, 301293, 819843, 2243058, 6167211, 17029473, 47200752, 131270283, 366195789, 1024380648, 2872770381, 8074967031, 22745832254, 64196912681, 181516532273, 514107418321, 1458407886019, 4143318012685
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 11*x^6 + 24*x^7 + 57*x^8 + 141*x^9 + 350*x^10 + 881*x^11 + 2267*x^12 + 5920*x^13 + 15601*x^14 + 41497*x^15 + ...
where A(x)^3 = A( x*A(x)^2/(1-x) )
and A(x) = x + x*(A(x) + A(x)^3 + A(x)^9 + A(x)^27 + ... A(x)^(3^n) + ...).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 33*x^7 + 84*x^8 + 208*x^9 + 522*x^10 + 1341*x^11 + 3479*x^12 + 9078*x^13 + 23907*x^14 + 63560*x^15 + ...
Let B(x) be the series reversion of A(x), B(A(x)) = x, then
B(x) = x - x^2 + x^3 - 2*x^4 + 3*x^5 - 4*x^6 + 6*x^7 - 9*x^8 + 13*x^9 - 20*x^10 + ... + (-1)^(n-1)*A078932(n-1)*x^n + ...
where x/B(x) = 1 + x + x^3 + x^9 + x^27 + x^81 + ... + x^(3^n) + ...
F(x) = A(x/(1+x)) = x + x^4 + 3*x^7 + 13*x^10 + 67*x^13 + 378*x^16 + 2253*x^19 + 13947*x^22 + 88803*x^25 + 577903*x^28 + 3826870*x^31 + 25703868*x^34 + ...
where F(x)^3 = F( x*F(x)^2/(1 - x*F(x)^2) )
and F(x) = x + x*(F(x)^3 + F(x)^9 + F(x)^27 + ... + F(x)^(3^n) + ...).
SPECIFIC VALUES.
A(t) = 2/3 at t = 0.3351780091733165997365854281871805851976265481916...
where 8/27 = A( (4/9)*t/(1-t) )
and t = (2/3)/(1 + Sum_{n>=0} (2/3)^(3^n)).
A(t) = 1/2 at t = 0.3073229277642929985518391822746766756418592443672...
where 1/8 = A( (1/4)*t/(1-t) )
and t = (1/2)/(1 + Sum_{n>=0} (1/2)^(3^n)).
A(1/3) = 0.640317989282342396539425948311398871030928082061168...
where A(1/3)^3 = A( A(1/3)^2/2 ).
A(1/4) = 0.347324237093006237340030053166266719890703533474663...
where A(1/4)^3 = A( A(1/4)^2/3 ).
A(1/5) = 0.254102848699628177600720471035831153854183353627930...
where A(1/5)^3 = A( A(1/5)^2/4 ).
A(1/10) = 0.111264157881789221767410282888976753122883279205707...
where A(1/10)^3 = A( A(1/10)^2/9 ).
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{a(n) = my(A = serreverse(x/(1 + sum(n=0,ceil(log(n+1)/log(3)), x^(3^n)) + x^3*O(x^n)) )); polcoeff(A,n)}
for(n=1, 40, print1(a(n), ", "))
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{a(n) = my(A=[1], Ax);
for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
A[#A] = -polcoeff( Ax^3 - subst(Ax, x, Ax^2*x/(1-x) ), #A+2) ); A[n]}
for(n=1, 40, print1(a(n), ", "))
A374570
Expansion of g.f. A(x) satisfying A(x)^2 = A( A(x)*C(x) ), where C(x) = x + C(x)^2 is the Catalan function (A000108).
Original entry on oeis.org
1, 1, 3, 8, 27, 90, 320, 1152, 4257, 15934, 60486, 231894, 897242, 3497638, 13725678, 54174286, 214923493, 856560918, 3427838222, 13768875142, 55494305328, 224359469870, 909656736876, 3697874061870, 15068978724200, 61545704828266, 251899370771284, 1033027441769384
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 90*x^6 + 320*x^7 + 1152*x^8 + 4257*x^9 + 15934*x^10 + 60486*x^11 + 231894*x^12 + ...
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 79*x^6 + 282*x^7 + 1046*x^8 + 3916*x^9 + 14907*x^10 + 57274*x^11 + 222194*x^12 + ...
where A(x)^2 = A( A(x)*C(x) ).
A(x)*C(x) = x^2 + 2*x^3 + 6*x^4 + 18*x^5 + 60*x^6 + 204*x^7 + 720*x^8 + 2586*x^9 + 9468*x^10 + 35124*x^11 + 131898*x^12 + ...
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n)*x^n + ,,,
where C(x) = (1 - sqrt(1 - 4*x))/2 is the Catalan function.
Let B(x) satisfy A(x*B(x)) = x, then
B(x) = 1 - x - x^2 + 2*x^3 - x^4 + x^5 + 2*x^6 - 6*x^7 - x^8 + 5*x^9 + x^10 + 2*x^12 - 8*x^13 - 6*x^14 + 22*x^15 - x^16 + ... + A374571(n)*x^n + ...
where C(x*B(x)) = x*B(x^2) and C(x) = x + C(x)^2.
Also notice that A(x-x^2) is the odd function starting as
A(x-x^2) = x + x^3 + 4*x^5 + 18*x^7 + 96*x^9 + 546*x^11 + 3274*x^13 + 20326*x^15 + 129622*x^17+ 843854*x^19 + ...
satisfying A(x-x^2)^2 = A( x*A(x-x^2) ).
SPECIFIC VALUES.
G.f. A(x) diverges at x = 1/4; what is the radius of convergence?
A(2/9) = 0.410501753930478190014767562028185186269192589705662553072...
where A(2/9)^2 = A( (1/3) * A(2/9) ).
A(1/5) = 0.307823207567908585715446000098072863270477544252476707540...
where A(1/5)^2 = A( A(1/5) * (1 - sqrt(1/5))/2 ).
A(1/6) = 0.222895676073964945442191376315546188067098171316653068516...
where A(1/6)^2 = A( A(1/6) * (1 - sqrt(1/3))/2 ).
A(1/8) = 0.149886223456626114071674919752683973970671151550604884301...
where A(1/8)^2 = A( A(1/8) * (1 - sqrt(1/2))/2 ).
A(1/10) = 0.11421035457722945538609562679806658343632346343476019471...
where A(1/10)^2 = A( A(1/10) * (1 - sqrt(3/5))/2 ).
A(1/12) = 0.09255115114959352826965125804331807348315032543228258146...
where A(1/12)^2 = A( A(1/12) * (1 - sqrt(2/3))/2 ).
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{a(n) = my(A=[1], Ax, C = serreverse(x-x^2 + x^2*O(x^n)));
for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
A[#A] = -polcoeff( Ax^2 - subst(Ax, x, Ax*C ), #A+1) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
A134527
G.f. A(x) satisfies: A(x) = Sum_{n>=0} [x*A(x)]^(2^n-1).
Original entry on oeis.org
1, 1, 1, 2, 5, 11, 24, 58, 149, 385, 1001, 2652, 7140, 19384, 52944, 145590, 402949, 1121117, 3133255, 8793372, 24774557, 70045871, 198672464, 565144064, 1611946284, 4609140916, 13209415116, 37937455636, 109171460104, 314736939884, 908930799572, 2629120466966
Offset: 0
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b:= proc(x, y, t) option remember; `if`(x<0 or y>x, 0,
`if`(x=0, 1, b(x-1, y+1, true)+`if`(t, add(
b(x-2^j+1, y-2^j+1, false), j=1..ilog2(y+1)), 0)))
end:
a:= n-> b(2*n, 0, true):
seq(a(n), n=0..32); # Alois P. Heinz, Apr 01 2019
-
f[x_, y_, d_] := f[x, y, d] = If[x < 0 || y < x, 0, If[x == 0 && y == 0, 1, f[x - 1, y, 0] + f[x, y - If[d == 0, 1, 2*d], If[d == 0, 1, 2*d]]]];Table[f[n, n, 0], {n, 0, 28}] (* David Scambler, May 07 2012 *)
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a(n)=polcoeff(serreverse(x/sum(j=0,#binary(n),x^(2^j-1)+ x*O(x^n))),n)
A374572
Expansion of g.f. A(x) satisfying A(x)^2 = A( x*(1+x)*A(x) ).
Original entry on oeis.org
1, 1, 1, 3, 5, 11, 27, 69, 183, 481, 1283, 3453, 9361, 25651, 70927, 197721, 555039, 1567345, 4449023, 12686465, 36323203, 104381397, 300958959, 870378337, 2524129349, 7338679127, 21386456807, 62459196233, 182776933033, 535861013939, 1573742036447, 4629306941913
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 5*x^5 + 11*x^6 + 27*x^7 + 69*x^8 + 183*x^9 + 481*x^10 + 1283*x^11 + 3453*x^12 + 9361*x^13 + 25651*x^14 + 70927*x^15 + ...
where A(x)^2 = A( x*(1+x)*A(x) ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 17*x^6 + 38*x^7 + 95*x^8 + 244*x^9 + 649*x^10 + 1738*x^11 + 4699*x^12 + ...
Let B(x) be the series reversion of g.f. A(x), B(A(x)) = x, then
B(x) = x - x^2 + x^3 - 3*x^4 + 9*x^5 - 25*x^6 + 71*x^7 - 219*x^8 + 693*x^9 - 2197*x^10 + 7069*x^11 - 23135*x^12 + ...
where B(x^2) = x*B(x)*(1 + B(x)).
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.301949314609828865985839329094529550482897401344979...
where 1/4 = A( t*(1 + t)/2 ).
A(3/10) = 0.492388112365452715229250795508017422919418907801551...
where A(3/10)^2 = A( (39/100)*A(3/10) ).
A(2/7) = 0.443877424659041232765055763766392304444609934055603...
where A(2/7)^2 = A( (18/49)*A(2/7) ).
A(1/4) = 0.352241294433584221893793757577235288109595399125986...
where A(1/4)^2 = A( (5/16)*A(1/4) ).
A(1/5) = 0.255826785620580342641277164817159026900345909888978...
where A(1/5)^2 = A( (6/25)*A(1/5) ).
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{a(n) = my(A=[0,1], Ax); for(i=1,n, A = concat(A,0); Ax = Ser(A);
A[#A] = polcoeff( subst(Ax,x, x*(1+x)*Ax ) - Ax^2, #A) );A[n+1]}
for(n=1,40,print1(a(n),", "))
A380554
G.f. A(x) satisfies A(x)^4 = A( A(x)^3 * x/(1-x) ).
Original entry on oeis.org
1, 1, 1, 1, 2, 6, 16, 36, 75, 163, 391, 991, 2498, 6150, 15016, 37116, 93482, 238154, 608074, 1551370, 3964200, 10176384, 26261500, 68034484, 176661828, 459534596, 1197777556, 3129475636, 8195867902, 21508247446, 56540427826, 148863643466, 392539322259, 1036662269875, 2741706892035
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 16*x^7 + 36*x^8 + 75*x^9 + 163*x^10 + 391*x^11 + 991*x^12 + 2498*x^13 + 6150*x^14 + 15016*x^15 + ...
where A(x)^4 = A( A(x)^3 * x/(1-x) );
also, A(x) = x*(1 + A(x) + A(x)^4 + A(x)^16 + A(x)^64 + ...).
RELATED SERIES.
x/(1 + x + x^4 + x^16 + x^64 + ...) = x - x^2 + x^3 - x^4 + x^6 - 2*x^7 + 3*x^8 - 3*x^9 + 2*x^10 - 3*x^12 + 6*x^13 - 8*x^14 + 8*x^15 - 5*x^16 + ...
where x = A( x/(1 + x + x^4 + x^16 + x^64 + ...) ).
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 42*x^8 + 112*x^9 + 288*x^10 + ...
A(x)^4 = x^4 + 4*x^5 + 10*x^6 + 20*x^7 + 39*x^8 + 88*x^9 + 228*x^10 + 600*x^11 + ...
SPECIFIC VALUES.
A(t) = 7/10 at t = 0.36018915820185609929548309671397017657231396...
where (7/10)^4 = A( (7/10)^3*t/(1-t) )
and t = (7/10)/(1 + Sum_{n>=0} (7/10)^(4^n)).
A(t) = 2/3 at t = 0.357324077294579321123715825007257976292387856...
where 16/81 = A( (8/27)*t/(1-t) )
and t = (2/3)/(1 + Sum_{n>=0} (2/3)^(4^n)).
A(t) = 1/2 at t = 0.319996875030517280093584464262123092506355813...
where 1/16 = A( (1/8)*t/(1-t) )
and t = (1/2)/(1 + Sum_{n>=0} (1/2)^(4^n)).
A(t) = 1/3 at t = 0.247706417742171319902767393551872977585317906...
where 1/81 = A( (1/27)*t/(1-t) )
and t = (1/3)/(1 + Sum_{n>=0} (1/3)^(4^n)).
A(1/3) = 0.54373202136840396341881074287828877295481851718413...
where A(1/3)^4 = A( A(1/3)^3*(1/2) ).
A(1/4) = 0.33766677567921691723942758840979376280294197783058...
where A(1/4)^4 = A( A(1/4)^3*(1/3) ).
A(1/5) = 0.25099215755350299738032710744403195608988446686839...
A(1/6) = 0.20032206620931060989695576481191496886558371212657...
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/* A(x) = Series_Reversion( x/(1 + Sum_{n>=0} x^(4^n)) ) */
{a(n) = my(A = serreverse( x/(1 + sum(m=0,ceil(log(n+1)/log(4)), x^(4^m) +x*O(x^n))) )); polcoef(A,n)}
for(n=1, 40, print1(a(n), ", "))
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/* A(x)^4 = A( A(x)^3 * x/(1-x) ) */
{a(n) = my(A=[1], Ax);
for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
A[#A] = -polcoeff( Ax^4 - subst(Ax, x, Ax^3*x/(1-x) ), #A+3) ); A[n]}
for(n=1, 40, print1(a(n), ", "))
A088704
a(n) equals the coefficient of x^n in f(x)^n where f(x)=1+sum(n>=0,x^(2^n)).
Original entry on oeis.org
1, 1, 3, 7, 23, 71, 231, 743, 2431, 7999, 26563, 88683, 297551, 1002015, 3385175, 11466887, 38933183, 132454719, 451423203, 1540920939, 5267257103, 18027478847, 61770328227, 211872505243, 727411948351, 2499560376671
Offset: 0
Showing 1-6 of 6 results.
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