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 1-10 of 10 results.

A360975 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(3*n) / n!.

Original entry on oeis.org

1, 2, 24, 476, 12380, 386220, 13821276, 552876504, 24318017424, 1162989779660, 59987353249560, 3316841570302680, 195648523068917828, 12263065689662763024, 814027053454694421000, 57059908657536257254704, 4212606733712173668180012, 326799750176069289173027820
Offset: 0

Views

Author

Paul D. Hanna, Feb 27 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 24*x^2 + 476*x^3 + 12380*x^4 + 386220*x^5 + 13821276*x^6 + 552876504*x^7 + 24318017424*x^8 + ... + a(n)*x^n + ...
where
A(x) = 1 + (d/dx x^2*A(x)^3) + (d^2/dx^2 x^4*A(x)^6)/2! + (d^3/dx^3 x^6*A(x)^9)/3! + (d^4/dx^4 x^8*A(x)^12)/4! + (d^5/dx^5 x^10*A(x)^15)/5! + (d^6/dx^6 x^12*A(x)^18)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^(3*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)^3), which begins
B(x) = x + x^2 + 8*x^3 + 119*x^4 + 2476*x^5 + 64370*x^6 + 1974468*x^7 + 69109563*x^8 + 2702001936*x^9 + ... + A360978(n)*x^n + ...
then A(x) = B'(x) and
B(x) = x * exp( x*A(x)^3 + (d/dx x^3*A(x)^6)/2! + (d^2/dx^2 x^5*A(x)^9)/3! + (d^3/dx^3 x^7*A(x)^12)/4! + (d^4/dx^4 x^9*A(x)^15)/5! + (d^5/dx^5 x^11*A(x)^18)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^(3*n))/n! + ... ).

Links

Crossrefs

Cf. A360950, A360973, A360974, A360978.

Programs

  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(2*m)*A^(3*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^2*A^3 +O(x^(n+2))))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)^3).
(3) B(x - x^2*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^(3*n) / n! ) is the g.f. of A360978.
(4) a(n) = (n+1) * A360978(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1/3)^n, where alfa = 2.598541481443... and c = 0.058191982295165... - Vaclav Kotesovec, Feb 28 2023
alfa = 7*LambertW(1/3) + 1/(1 + LambertW(1/3)). - Vaclav Kotesovec, Mar 13 2023

A360976 G.f. satisfies: A(x) = Series_Reversion(x - x^3*A'(x)).

Original entry on oeis.org

1, 1, 6, 66, 1027, 20274, 479403, 13118703, 406181493, 14007078204, 531778565544, 22028404578840, 988535991793203, 47773653611710429, 2473958531200630992, 136684964338470273828, 8026375457238402039978, 499251236257852169668461, 32794618460003080060574283
Offset: 1

Views

Author

Paul D. Hanna, Feb 27 2023

Keywords

Comments

a(n) = A360973(n-1)/(2*n-1) for n >= 1.

Examples

			G.f.: A(x) = x + x^3 + 6*x^5 + 66*x^7 + 1027*x^9 + 20274*x^11 + 479403*x^13 + 13118703*x^15 + 406181493*x^17 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)) = x, where
A'(x) = 1 + 3*x^2 + 30*x^4 + 462*x^6 + 9243*x^8 + 223014*x^10 + 6232239*x^12 + 196780545*x^14 + ... + A360973(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)) + (d^2/dx^2 x^6*A'(x)^2)/2! + (d^3/dx^3 x^9*A'(x)^3)/3! + (d^4/dx^4 x^12*A'(x)^4)/4! + (d^5/dx^5 x^15*A'(x)^5)/5! + (d^6/dx^6 x^18*A'(x)^6)/6! + ... + (d^n/dx^n x^(3*n)*A'(x)^n)/n! + ...
Further,
A(x) = x * exp( x^2*A'(x) + (d/dx x^5*A'(x)^2)/2! + (d^2/dx^2 x^8*A'(x)^3)/3! + (d^3/dx^3 x^11*A'(x)^4)/4! + (d^4/dx^4 x^14*A'(x)^5)/5! + (d^5/dx^5 x^17*A'(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^n)/n! + ... ).

Links

Crossrefs

Cf. A360973, A229619, A360977, A360978.

Programs

  • PARI
    {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^3*A'+x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^3*A'(x)).
(2) A(x) = x + A(x)^3 * A'(A(x)).
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^n / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^n / n!.

A360977 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)^2).

Original entry on oeis.org

1, 1, 6, 65, 978, 18354, 406396, 10274553, 290411050, 9045087486, 307195803380, 11287142857810, 445880114920868, 18841715641048244, 848104833677384856, 40515633401768290017, 2047629885363936119690, 109168898899192961652766, 6124201250273921324876116
Offset: 1

Views

Author

Paul D. Hanna, Feb 27 2023

Keywords

Comments

a(n) = A360974(n-1)/n for n >= 1.

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 65*x^4 + 978*x^5 + 18354*x^6 + 406396*x^7 + 10274553*x^8 + 290411050*x^9 + ...
By definition, A(x - x^2*A'(x)^2) = x, where
A'(x) = 1 + 2*x + 18*x^2 + 260*x^3 + 4890*x^4 + 110124*x^5 + 2844772*x^6 + 82196424*x^7 + ... + A360974(n)*x^n + ...
Also,
A'(x) = 1 + (d/dx x^2*A'(x)^2) + (d^2/dx^2 x^4*A'(x)^4)/2! + (d^3/dx^3 x^6*A'(x)^6)/3! + (d^4/dx^4 x^8*A'(x)^8)/4! + (d^5/dx^5 x^10*A'(x)^10)/5! + ... + (d^n/dx^n x^(2*n)*A'(x)^(2*n))/n! + ...
Further,
A(x) = x * exp( x*A'(x)^2 + (d/dx x^3*A'(x)^4)/2! + (d^2/dx^2 x^5*A'(x)^6)/3! + (d^3/dx^3 x^7*A'(x)^8)/4! + (d^4/dx^4 x^9*A'(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A'(x)^(2*n))/n! + ... ).

Links

Crossrefs

Cf. A360974, A229619, A360976, A360978.

Programs

  • PARI
    {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^2*A'^2 +x*O(x^(n+1)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n may be defined by the following.
(1) A(x) = Series_Reversion(x - x^2*A'(x)^2).
(2) A(x) = x + A(x)^2 * A'(A(x))^2.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A'(x)^(2*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A'(x)^(2*n) / n!.

A361302 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^3*A'(x)^3).

Original entry on oeis.org

1, 1, 12, 291, 10243, 460632, 24830853, 1546531419, 108716955930, 8489321379453, 727903248520260, 67935651633100242, 6853940772480079902, 743261410711529857459, 86224073603509482578211, 10656471864208782754351131, 1398062659621217619155428209
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^3 + 12*x^5 + 291*x^7 + 10243*x^9 + 460632*x^11 + 24830853*x^13 + 1546531419*x^15 + 108716955930*x^17 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^3) = x, where
A'(x) = 1 + 3*x^2 + 60*x^4 + 2037*x^6 + 92187*x^8 + 5066952*x^10 + 322801089*x^12 + 23197971285*x^14 + ... + A361536(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)^3) + (d^2/dx^2 x^6*A'(x)^6)/2! + (d^3/dx^3 x^9*A'(x)^9)/3! + (d^4/dx^4 x^12*A'(x)^12)/4! + (d^5/dx^5 x^15*A'(x)^15)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(3*n))/n! + ...
Further,
A(x) = x * exp( x^2*A'(x)^3 + (d/dx x^5*A'(x)^6)/2! + (d^2/dx^2 x^8*A'(x)^9)/3! + (d^3/dx^3 x^11*A'(x)^12)/4! + (d^4/dx^4 x^14*A'(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(3*n))/n! + ... ).

Links

Crossrefs

Cf. A361536.
Cf. A229619, A360976, A360977, A360978, A361307, A361308, A361309, A361310, A361311.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^3*A'^3 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^3*A'(x)^3).
(2) A(x) = x + A(x)^3 * A'(A(x))^3.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(3*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(3*n) / n! is the g.f. of A361536.
(5) a(n) = A361536(n-1)/(2*n-1) for n >= 1.

A361307 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^3*A'(x)^4).

Original entry on oeis.org

1, 1, 15, 462, 20719, 1187628, 81575478, 6470236914, 578865763791, 57491440616067, 6266161502595672, 743009082083639748, 95191896469891628934, 13103364445591714775407, 1928820020328686200102278, 302383969785427961077318020, 50307405653295945234562827135
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^3 + 15*x^5 + 462*x^7 + 20719*x^9 + 1187628*x^11 + 81575478*x^13 + 6470236914*x^15 + 578865763791*x^17 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^4) = x, where
A'(x) = 1 + 3*x^2 + 75*x^4 + 3234*x^6 + 186471*x^8 + 13063908*x^10 + 1060481214*x^12 + 97053553710*x^14 + ... + A361537(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)^4) + (d^2/dx^2 x^6*A'(x)^8)/2! + (d^3/dx^3 x^9*A'(x)^12)/3! + (d^4/dx^4 x^12*A'(x)^16)/4! + (d^5/dx^5 x^15*A'(x)^20)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(4*n))/n! + ...
Further,
A(x) = x * exp( x^2*A'(x)^4 + (d/dx x^5*A'(x)^8)/2! + (d^2/dx^2 x^8*A'(x)^12)/3! + (d^3/dx^3 x^11*A'(x)^16)/4! + (d^4/dx^4 x^14*A'(x)^20)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(4*n))/n! + ... ).

Links

Crossrefs

Cf. A361537.
Cf. A229619, A360976, A360977, A360978, A361302, A361308, A361309, A361310, A361311.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^3*A'^4 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^3*A'(x)^4).
(2) A(x) = x + A(x)^3 * A'(A(x))^4.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(4*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(4*n) / n! is the g.f. of A361537.
(5) a(n) = A361537(n-1)/(2*n-1) for n >= 1.

A361308 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)).

Original entry on oeis.org

1, 1, 8, 122, 2676, 75197, 2548336, 100461956, 4500071172, 225305924896, 12456434569184, 753380353835754, 49473301917640864, 3505613955205438686, 266627715169575108168, 21667902182055638829520, 1873978995774161192935320, 171874439346918445003163152
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^4 + 8*x^7 + 122*x^10 + 2676*x^13 + 75197*x^16 + 2548336*x^19 + 100461956*x^22 + 4500071172*x^25 + ... + a(n)*x^(3*n-2) + ...
By definition, A(x - x^4*A'(x)) = x, where
A'(x) = 1 + 4*x^3 + 56*x^6 + 1220*x^9 + 34788*x^12 + 1203152*x^15 + 48418384*x^18 + 2210163032*x^21 + ... + A361541(n)*x^(3*n) + ...
Also,
A'(x) = 1 + (d/dx x^4*A'(x)) + (d^2/dx^2 x^8*A'(x)^2)/2! + (d^3/dx^3 x^12*A'(x)^3)/3! + (d^4/dx^4 x^16*A'(x)^4)/4! + (d^5/dx^5 x^20*A'(x)^5)/5! + ... + (d^n/dx^n x^(4*n)*A'(x)^n)/n! + ...
Further,
A(x) = x * exp( x^3*A'(x) + (d/dx x^7*A'(x)^2)/2! + (d^2/dx^2 x^11*A'(x)^3)/3! + (d^3/dx^3 x^15*A'(x)^4)/4! + (d^4/dx^4 x^19*A'(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A'(x)^n)/n! + ... ).

Links

Crossrefs

Cf. A361541.
Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361309, A361310, A361311.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^4*A' +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^4*A'(x)).
(2) A(x) = x + A(x)^4 * A'(A(x)).
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^n / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^n / n! is the g.f. of A361541.
(5) a(n) = A361541(n-1)/(3*n-2) for n >= 1.

A361309 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)^2).

Original entry on oeis.org

1, 1, 12, 294, 10556, 488105, 27237748, 1766404068, 129955274460, 10668008963012, 965419570076880, 95430263520948342, 10228351567332536636, 1181548204752647642190, 146354418172125510269224, 19353257235976807395819160, 2721549078621826864159594548
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^4 + 12*x^7 + 294*x^10 + 10556*x^13 + 488105*x^16 + 27237748*x^19 + 1766404068*x^22 + 129955274460*x^25 + ... + a(n)*x^(3*n-2) + ...
By definition, A(x - x^4*A'(x)^2) = x, where
A'(x) = 1 + 4*x^3 + 84*x^6 + 2940*x^9 + 137228*x^12 + 7809680*x^15 + 517517212*x^18 + 38860889496*x^21 + ... + A361542(n)*x^(3*n) + ...
Also,
A'(x) = 1 + (d/dx x^4*A'(x)^2) + (d^2/dx^2 x^8*A'(x)^4)/2! + (d^3/dx^3 x^12*A'(x)^6)/3! + (d^4/dx^4 x^16*A'(x)^8)/4! + (d^5/dx^5 x^20*A'(x)^10)/5! + ... + (d^n/dx^n x^(4*n)*A'(x)^(2*n))/n! + ...
Further,
A(x) = x * exp( x^3*A'(x)^2 + (d/dx x^7*A'(x)^4)/2! + (d^2/dx^2 x^11*A'(x)^6)/3! + (d^3/dx^3 x^15*A'(x)^8)/4! + (d^4/dx^4 x^19*A'(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A'(x)^(2*n))/n! + ... ).

Links

Crossrefs

Cf. A361542.
Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361310, A361311.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^4*A'^2 +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^4*A'(x)^2).
(2) A(x) = x + A(x)^4 * A'(A(x))^2.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^(2*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^(2*n) / n! is the g.f. of A361542.
(5) a(n) = A361542(n-1)/(3*n-2) for n >= 1.

A361310 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)^3).

Original entry on oeis.org

1, 1, 16, 538, 26676, 1705373, 131524408, 11778395196, 1195433981028, 135247561603456, 16853285080609312, 2292048750536003426, 337754031605269049112, 53608164572529006153454, 9118712400086550140230888, 1655104918901340697851158384, 319341008921919836189242604080
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^4 + 16*x^7 + 538*x^10 + 26676*x^13 + 1705373*x^16 + 131524408*x^19 + 11778395196*x^22 + ... + a(n)*x^(3*n-2) + ...
By definition, A(x - x^4*A'(x)^3) = x, where
A'(x) = 1 + 4*x^3 + 112*x^6 + 5380*x^9 + 346788*x^12 + 27285968*x^15 + 2498963752*x^18 + 259124694312*x^21 + ... + A361543(n)*x^(3*n) + ...
Also,
A'(x) = 1 + (d/dx x^4*A'(x)^3) + (d^2/dx^2 x^8*A'(x)^6)/2! + (d^3/dx^3 x^12*A'(x)^9)/3! + (d^4/dx^4 x^16*A'(x)^12)/4! + (d^5/dx^5 x^20*A'(x)^15)/5! + ... + (d^n/dx^n x^(4*n)*A'(x)^(3*n))/n! + ...
Further,
A(x) = x * exp( x^3*A'(x)^3 + (d/dx x^7*A'(x)^6)/2! + (d^2/dx^2 x^11*A'(x)^9)/3! + (d^3/dx^3 x^15*A'(x)^12)/4! + (d^4/dx^4 x^19*A'(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A'(x)^(3*n))/n! + ... ).

Links

Crossrefs

Cf. A361543.
Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361309, A361311.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^4*A'^3 +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^4*A'(x)^3).
(2) A(x) = x + A(x)^4 * A'(A(x))^3.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^(3*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^(3*n) / n! is the g.f. of A361543.
(5) a(n) = A361543(n-1)/(3*n-2) for n >= 1.

A361311 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^5*A'(x)).

Original entry on oeis.org

1, 1, 10, 195, 5520, 201255, 8881551, 457227585, 26805712005, 1759840463070, 127784731466660, 10164274303786460, 878859905526721250, 82080454974318915935, 8235485665033295289810, 883569144560890419421630, 100952601749463417250801935, 12239031817482031919864850550
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^5 + 10*x^9 + 195*x^13 + 5520*x^17 + 201255*x^21 + 8881551*x^25 + 457227585*x^29 + ... + a(n)*x^(4*n-3) + ...
By definition, A(x - x^5*A'(x)) = x, where
A'(x) = 1 + 5*x^4 + 90*x^8 + 2535*x^12 + 93840*x^16 + 4226355*x^20 + 222038775*x^24 + ... + A361551(n)*x^(4*n) + ...
Also,
A'(x) = 1 + (d/dx x^5*A'(x)) + (d^2/dx^2 x^10*A'(x)^2)/2! + (d^3/dx^3 x^15*A'(x)^3)/3! + (d^4/dx^4 x^20*A'(x)^4)/4! + (d^5/dx^5 x^25*A'(x)^5)/5! + ... + (d^n/dx^n x^(5*n)*A'(x)^n)/n! + ...
Further,
A(x) = x * exp( x^4*A'(x) + (d/dx x^9*A'(x)^2)/2! + (d^2/dx^2 x^14*A'(x)^3)/3! + (d^3/dx^3 x^19*A'(x)^4)/4! + (d^4/dx^4 x^24*A'(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(5*n-1)*A'(x)^n)/n! + ... ).

Links

Crossrefs

Cf. A361551.
Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361309, A361310.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^5*A' +x*O(x^(4*n)))); polcoeff(A, 4*n-3)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(4*n-3) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^5*A'(x)).
(2) A(x) = x + A(x)^5 * A'(A(x)).
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(5*n-1) * A'(x)^n / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(5*n) * A'(x)^n / n! is the g.f. of A361551.
(5) a(n) = A361551(n-1)/(4*n-3) for n >= 1.

A361047 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion(x - x^3*A'(x)^2).

Original entry on oeis.org

1, 1, 9, 159, 4051, 131688, 5132793, 231332589, 11778989157, 666865748751, 41494745678544, 2812781975630049, 206264308294757115, 16268935714201604701, 1373512281722006688063, 123601628009085259269819, 11812339040349301277253801, 1194940136210629914238593762
Offset: 1

Views

Author

Paul D. Hanna, Mar 03 2023

Keywords

Comments

Conjecture: a(n) == 1 (mod 3) iff n = (3^k - 1)/2 for k >= 0, otherwise a(n) == 0 (mod 3).

Examples

			G.f.: A(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^2) = x, where
A'(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + A361046(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)^2) + (d^2/dx^2 x^6*A'(x)^4)/2! + (d^3/dx^3 x^9*A'(x)^6)/3! + (d^4/dx^4 x^12*A'(x)^8)/4! + (d^5/dx^5 x^15*A'(x)^10)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(2*n))/n! + ...
Further,
A(x) = x * exp( x^2*A'(x)^2 + (d/dx x^5*A'(x)^4)/2! + (d^2/dx^2 x^8*A'(x)^6)/3! + (d^3/dx^3 x^11*A'(x)^8)/4! + (d^4/dx^4 x^14*A'(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(2*n))/n! + ... ).

Links

Crossrefs

Cf. A361046, A229619, A360977, A360978.

Programs

  • PARI
    {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^3*(A')^2 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^3*A'(x)^2).
(2) A(x) = x + A(x)^3 * A'(A(x))^2.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(2*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(2*n) / n!, where A'(x) is the g.f. of A361046.
a(n) = A361046(n-1)/(2*n-1) for n >= 1.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.