cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A372530 Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x)) ).

Original entry on oeis.org

1, 1, 3, 9, 33, 125, 501, 2065, 8739, 37685, 165107, 732681, 3286679, 14878885, 67889851, 311896993, 1441536321, 6698017445, 31269529601, 146601334841, 689945263873, 3258334336349, 15436401872405, 73341269533009, 349381321611505, 1668434132560765, 7985390073708765
Offset: 1

Views

Author

Paul D. Hanna, May 13 2024

Keywords

Comments

Compare to the following identities of the Catalan function C(x) = x + C(x)^2 (A000108):
(1) C(x)^2 = C( x*C(x)*(1 + C(x)) ),
(2) C(x)^4 = C( x*C(x)^3*(1 + C(x))*(1 + C(x)^2) ),
(3) C(x)^8 = C( x*C(x)^7*(1 + C(x))*(1 + C(x)^2)*(1 + C(x)^4) ),
(4) C(x)^(2^n) = C( x*C(x)^(2^n-1)*Product_{k=0..n-1} (1 + C(x)^(2^k)) ) for n > 0.

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 9*x^4 + 33*x^5 + 125*x^6 + 501*x^7 + 2065*x^8 + 8739*x^9 + 37685*x^10 + 165107*x^11 + 732681*x^12 + ...
where A( x*A(x)/(1 - A(x)) ) = A(x)^2.
RELATED SERIES.
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)) = x - x^2 - x^3 + x^4 - x^5 + x^6 + x^7 - x^8 - x^9 + x^10 + x^11 - x^12 + x^13 - x^14 - x^15 + x^16 + ... + (-1)^A010060(n-1) * x^n + ...
thus,
x = A(x) * (1 - A(x)) * (1 - A(x)^2) * (1 - A(x)^4) * (1 - A(x)^8) * (1 - A(x)^16) * ... * (1 - A(x)^(2^n)) * ...
SPECIFIC VALUES.
A(t) = 1/3 at t = (1/3) * Product_{n>=0} (1 - 1/3^(2^n)) = 0.195062471888103139123433255203480726664398592...
A(t) = 1/4 at t = (1/4) * Product_{n>=0} (1 - 1/4^(2^n)) = 0.175091932719784804433277263483089433821043251...
A(1/6) = 0.2285942310240955503097133963953487564542629539800372181...
A(1/7) = 0.1803372891149269875688065840927292319030238580575714990...
A(1/8) = 0.1506715662175837437127190414569072051853697889895576799...
A(1/6)^2 = A(t) at t = (1/6)*A(1/6)/(1 - A(1/6)) = 0.0493891023845...
A(1/7)^2 = A(t) at t = (1/7)*A(1/7)/(1 - A(1/7)) = 0.0314305744685...

Links

Crossrefs

Cf. A373312, A373313, A372531, A371713, A371709, A010060.

Programs

  • PARI
    {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 - F) ) - F^2, #A) ); H=A; A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x*A(x)/(1 - A(x)) ).
(2) A(x)^4 = A( x*A(x)^3/((1 - A(x))*(1 - A(x)^2)) ).
(3) A(x)^8 = A( x*A(x)^7/((1 - A(x))*(1 - A(x)^2)*(1 - A(x)^4)) ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1)/Product_{k=0..n-1} (1 - A(x)^(2^k)) ) for n > 0.
(5) A(x) = x / Product_{n>=0} (1 - A(x)^(2^n)).
(6) A(x) = x * Product_{n>=0} (1 + A(x)^(2^n))^(n+1). - Paul D. Hanna, Jun 26 2024
(7) A(x) = Series_Reversion( x * Product_{n>=0} (1 - x^(2^n)) ).
(8) x = Sum_{n>=1} (-1)^A010060(n-1) * A(x)^n, where A010060 is the Thue-Morse sequence.
The radius of convergence r and A(r) satisfy 1 = Sum_{n>=0} 2^n * A(r)^(2^n)/(1 - A(r)^(2^n)) and r = A(r) * Product_{n>=0} (1 - A(r)^(2^n)), where r = 0.19736158352631556925015099049581233030702919287488... and A(r) = 0.37298513723316144189484491702105095014110332846051...
Given r and A(r) above, A(r) also satisfies 1 = Sum_{n>=0} (n+1)*2^n * A(r)^(2^n)/(1 + A(r)^(2^n)). - Paul D. Hanna, Jun 26 2024

A371716 Expansion of g.f. A(x) satisfies A( x*A(x)^3 + x*A(x)^4 ) = A(x)^4.

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 22, 57, 131, 298, 738, 2003, 5600, 15380, 41224, 109769, 296010, 813333, 2261818, 6307070, 17560050, 48877852, 136457322, 382803675, 1078562370, 3047295816, 8623046992, 24432992884, 69345396556, 197211214852, 561975160288, 1604186098089, 4585779820379
Offset: 1

Views

Author

Paul D. Hanna, May 03 2024

Keywords

Comments

Compare to the following identities of the Catalan function C(x) = x + C(x)^2 (A000108):
(1) C(x)^2 = C( x*C(x)*(1 + C(x)) ),
(2) C(x)^4 = C( x*C(x)^3*(1 + C(x))*(1 + C(x)^2) ),
(3) C(x)^8 = C( x*C(x)^7*(1 + C(x))*(1 + C(x)^2)*(1 + C(x)^4) ),
(4) C(x)^(2^n) = C( x*C(x)^(2^n-1)*Product_{k=0..n-1} (1 + C(x)^(2^k)) ) for n > 0.

Examples

			G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 7*x^6 + 22*x^7 + 57*x^8 + 131*x^9 + 298*x^10 + 738*x^11 + 2003*x^12 + 5600*x^13 + 15380*x^14 + ...
where A( x*A(x)^3*(1 + A(x)) ) = A(x)^4.
RELATED SERIES.
Let B(x) be the series reversion of g.f. A(x), B(A(x)) = x, then
B(x) = x/((1+x)*(1+x^4)*(1+x^16)*(1+x^64)*(1+x^256)*(1+x^1024)*...) = x - x^2 + x^3 - x^4 + x^9 - x^10 + x^11 - x^12 + x^33 - x^34 + ...
We can show that g.f. A(x) = A( x*A(x)^3*(1 + A(x)) )^(1/4) satisfies
(4) A(x) = x * Product_{n>=0} (1 + A(x)^(4^n))
by substituting x*A(x)^3*(1 + A(x)) for x in (4) to obtain
A(x)^4 = x * A(x)^3*(1 + A(x)) * Product_{n>=1} (1 + A(x)^(4^n))
which is equivalent to formula (4).
SPECIFIC VALUES.
A(1/3) = 0.6209428791888803994421374991623399343094...
A(1/4) = 0.3392462304609640143453810140211726768116...
A(1/5) = 0.2512464727722296135954631316870173555867...
A(t) = 1/2 and A(t*3/16) = 1/16 at t = 0.31372070319804379323613829910755157...
A(t) = 1/3 and A(t*4/81) = 1/81 at t = 0.24695121377537689193140239461709572...
A(t) = 1/4 and A(t*5/256) = 1/256 at t = 0.199221789836883544932674834867379...

Links

Crossrefs

Cf. A371709, A371713.

Programs

  • PARI
    /* Using series reversion of x/Product_{n>=0} (1 + x^(4^n)) */
{a(n) = my(A); A = serreverse( x/prod(k=0, ceil(log(n)/log(4)), (1 + x^(4^k) +x*O(x^n)) ) ); polcoeff(A, n)}
for(n=1, 35, print1(a(n), ", "))
  • PARI
    /* Using A(x)^4 = A( x*A(x)^3 + x*A(x)^4 ) */
{a(n) = my(A=[1], F); for(i=1, n, A = concat(A, 0); F = x*Ser(A);
A[#A] = polcoeff( subst(F, x, x*F^3 + x*F^4 ) - F^4, #A+3) ); A[n]}
for(n=1, 35, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^4 = A( x*A(x)^3*(1 + A(x)) ).
(2) A(x)^16 = A( x*A(x)^15*(1 + A(x))*(1 + A(x)^4) ).
(3) A(x)^64 = A( x*A(x)^63*(1 + A(x))*(1 + A(x)^4)*(1 + A(x)^16) ).
(4) A(x) = x * Product_{n>=0} (1 + A(x)^(4^n)).
(5) A(x) = Series_Reversion( x / Product_{n>=0} (1 + x^(4^n)) ).
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 4^n * A(r)^(4^n) / (1 + A(r)^(4^n)) and r = A(r) / Product_{n>=0} (1 + A(r)^(4^n)), where r = 0.33394799468036632700505690802809657984166722... and A(r) = 0.64588119033501052326223671937159514208118071...

A372578 Expansion of g.f. A(x) satisfying A( x*A(x) + 2*A(x)^3 ) = A(x)^2.

Original entry on oeis.org

1, 2, 10, 60, 406, 2940, 22304, 174960, 1407582, 11550396, 96299472, 813433712, 6946442776, 59872428672, 520174647424, 4550665293920, 40052871669422, 354421196057404, 3151211548631856, 28137903707808048, 252219507331523688, 2268719274696321856, 20472066335198022080, 185268984285773695200
Offset: 1

Views

Author

Paul D. Hanna, Jun 17 2024

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 10*x^3 + 60*x^4 + 406*x^5 + 2940*x^6 + 22304*x^7 + 174960*x^8 + 1407582*x^9 + 11550396*x^10 + 96299472*x^11 + 813433712*x^12 + ...
where A( x*A(x) + 2*A(x)^3 ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 160*x^5 + 1152*x^6 + 8704*x^7 + 68088*x^8 + 546656*x^9 + 4478720*x^10 + 37294080*x^11 + ...
A(x)^3 = x^3 + 6*x^4 + 42*x^5 + 308*x^6 + 2358*x^7 + 18612*x^8 + 150424*x^9 + 1238688*x^10 + 10355982*x^11 + 87672468*x^12 + ...
The series reversion R(x) of A(x), R(A(x)) = x, begins:
R(x) = x - 2*x^2 - 2*x^3 - 2*x^5 - 2*x^9 - 2*x^17 - 2*x^33 - 2*x^65 - 2*x^129 - 2*x^257 - 2*x^513 + ... + -2*x^(2^n+1) + ...
SPECIFIC VALUES.
A(1/10) = 0.1580645870348513671680526916072548213169829162556439...
A(1/11) = 0.1278454819475039498675733418966788971517121949516108...
A(1/12) = 0.1104694875320629136831876267359845627848091250498995...

Links

Crossrefs

Cf. A371713, A356782.

Programs

  • PARI
    {a(n) = my(A=serreverse(x - 2*x*sum(m=0,#binary(n),x^(2^m) +x*O(x^n)))); polcoeff(A,n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {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 + 2*F^3) - F^2, #A) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x*A(x) + 2*A(x)^3 ).
(2) A(x)^4 = A( x*A(x)^3 + 2*A(x)^5 + 2*A(x)^6 ).
(3) A(x)^8 = A( x*A(x)^7 + 2*A(x)^9 + 2*A(x)^10 + 2*A(x)^12 ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1) + 2*Sum_{k=0,n-1} A(x)^(2^n+2^k) ).
(5) A(x) = x + 2*Sum_{n>=0} A(x)^(2^n+1).
(6) A(x) = Series_Reversion( x - 2*x*Sum_{n>=0} x^(2^n) ).
The radius of convergence r and A(r) satisfy: 1 = Sum_{n>=0} 2*(2^n+1) * A(r)^(2^n) and r = A(r) - 2*Sum_{n>=0} A(r)^(2^n+1), where r = 0.103594274393575546296984777950632418580281502255382627835... and A(r) = 0.191573759982214348953869719011237665707785580853712880852...

A372526 Expansion of g.f. A(x) satisfying A( x*A(x)^2 + A(x)^4 ) = A(x)^3.

Original entry on oeis.org

1, 1, 2, 6, 20, 70, 256, 969, 3762, 14895, 59916, 244179, 1006026, 4183396, 17534888, 74007851, 314256048, 1341575769, 5754629794, 24789907450, 107202369386, 465209278326, 2025212712660, 8842042378050, 38707067608872, 169860383434800, 747096961093560, 3292855742992644
Offset: 1

Views

Author

Paul D. Hanna, May 04 2024

Keywords

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 256*x^7 + 969*x^8 + 3762*x^9 + 14895*x^10 + 59916*x^11 + 244179*x^12 + ...
where A( x*A(x)^2 + A(x)^4 ) = A(x)^3.
RELATED SERIES.
(1) A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 432*x^8 + 1676*x^9 + 6633*x^10 + 26676*x^11 + 108696*x^12 + ...
(2) x*A(x)^2 + A(x)^4 = x^3 + 3*x^4 + 9*x^5 + 30*x^6 + 108*x^7 + 405*x^8 + 1560*x^9 + 6138*x^10 + 24570*x^11 + 99738*x^12 + ...
(3) Let R(x) be the series reversion of A(x), R(A(x)) = x, then
R(x) = x - x^2 - x^4 - x^10 - x^28 - x^82 - x^244 - x^730 + ... + -x^(3^n+1) + ...
SPECIFIC VALUES.
A(1/5) = 0.2937167157779136500722875625899113632023...
A(1/6) = 0.2150539986528250703029216090552606059919...
A(1/7) = 0.1740789503092637057579787813575613522976...
A(1/8) = 0.1471095742959948638409574049543396207684...

Links

Crossrefs

Cf. A371713, A371716.

Programs

  • PARI
    /* Using series reversion of x - x*Sum_{n>=0} x^(3^n) */
{a(n) = my(A); A = serreverse( x - x*sum(k=0, ceil(log(n)/log(3)), x^(3^k) +x*O(x^n)) ); polcoeff(A, n)}
for(n=1, 35, print1(a(n), ", "))
  • PARI
    /* Using A(x)^3 = A( x*A(x)^2 + A(x)^4 ) */
{a(n) = my(A=[1], F); for(i=1, n, A = concat(A, 0); F = x*Ser(A);
A[#A] = polcoeff( subst(F, x, x*F^2 + F^4 ) - F^3, #A+2) ); A[n]}
for(n=1, 35, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( x*A(x)^2 + A(x)^4 ).
(2) A(x)^9 = A( x*A(x)^8 + A(x)^10 + A(x)^12 ).
(3) A(x)^27 = A( x*A(x)^26 + A(x)^28 + A(x)^30 + A(x)^36 ).
(4) A(x)^(3^n) = A( x*A(x)^(3^n-1) + Sum_{k=0..n-1} A(x)^(3^n+3^k) ) for n > 0.
(5) A(x) = x + Sum_{n>=0} A(x)^(3^n+1).
(6) A(x) = Series_Reversion(x - x*Sum_{n>=0} x^(3^n) ).
The radius of convergence r and A(r) satisfy: 1 = Sum_{n>=0} (3^n+1) * A(r)^(3^n) and r = A(r) - Sum_{n>=0} A(r)^(3^n+1), where r = 0.214732801800375010254079407876131682823903064701286670006... and A(r) = 0.384967312289976324530970877165834568783164468488676531438...
Showing 1-4 of 4 results.

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