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 14 results. Next

A360950 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^n / n!.

Original entry on oeis.org

1, 2, 12, 108, 1240, 16932, 264740, 4631320, 89270316, 1875586380, 42610756408, 1040307155304, 27157913296228, 754950111249488, 22267948484559720, 694746226969477744, 22863695087986373968, 791675941860401322852, 28776089467457429038620, 1095679176790207081120360
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 108*x^3 + 1240*x^4 + 16932*x^5 + 264740*x^6 + 4631320*x^7 + 89270316*x^8 + 1875586380*x^9 + ...
where
A(x) = 1 + (d/dx x^2*A(x)) + (d^2/dx^2 x^4*A(x)^2)/2! + (d^3/dx^3 x^6*A(x)^3)/3! + (d^4/dx^4 x^8*A(x)^4)/4! + (d^5/dx^5 x^10*A(x)^5)/5! + (d^6/dx^6 x^12*A(x)^6)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^n)/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)), which begins
B(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2822*x^6 + 37820*x^7 + 578915*x^8 + 9918924*x^9 + 187558638*x^10 + ... + A229619(n)*x^n + ...
then A(x) = B'(x) and
B(x) = x * exp( x*A(x) + (d/dx x^3*A(x)^2)/2! + (d^2/dx^2 x^5*A(x)^3)/3! + (d^3/dx^3 x^7*A(x)^4)/4! + (d^4/dx^4 x^9*A(x)^5)/5! + (d^5/dx^5 x^11*A(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n)/n! + ... ).

Links

Crossrefs

Cf. A229619, A356848, A088714, A303063.

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^m/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^n / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)).
(3) B(x - x^2*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^n / n! ) is the g.f. of A229619.
(4) a(n) = (n+1) * A229619(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1)^n, where alfa = 3*LambertW(1) + 1/(1 + LambertW(1)) = 2.33953361459... and c = 0.1926079501120681239... - Vaclav Kotesovec, Feb 27 2023

A360973 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^n / n!.

Original entry on oeis.org

1, 3, 30, 462, 9243, 223014, 6232239, 196780545, 6905085381, 266134485876, 11167349876424, 506653305313320, 24713399794830075, 1289888647516181583, 71744797404818298768, 4237233894492578488668, 264870390088867267319274, 17473793269024825938396135
Offset: 0

Views

Author

Paul D. Hanna, Feb 27 2023

Keywords

Examples

			G.f.: A(x) = 1 + 3*x^2 + 30*x^4 + 462*x^6 + 9243*x^8 + 223014*x^10 + 6232239*x^12 + 196780545*x^14 + 6905085381*x^16 + ... + a(n)*x^(2*n) + ...
where
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! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)), which begins
B(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 + ... + A360976(n)*x^(2*n-1) + ...
then A(x) = B'(x) and
B(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. A360950, A360974, A360975, A360976.
Cf. A229619, A356848, A088714, A303063.

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^(3*m)*A^m/m!)) +O(x^(2*n+1))); polcoeff(A, 2*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^3*A +O(x^(2*n+3))))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^n / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)).
(3) B(x - x^3*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^n / n! ) is the g.f. of A360976.
(4) a(n) = (2*n+1) * A360976(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa * 2^n / LambertW(1)^n, where alfa = 1.50465021094584... and c = 0.36053267424501... - Vaclav Kotesovec, Feb 28 2023
alfa = (9*LambertW(1) - 1 + 3/(1 + LambertW(1)))/4. - Vaclav Kotesovec, Mar 13 2023

A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.

Original entry on oeis.org

1, 2, 18, 260, 4890, 110124, 2844772, 82196424, 2613699450, 90450874860, 3379153837180, 135445714293720, 5796441493971284, 263784018974675416, 12721572505160772840, 648250134428292640272, 34809708051186914034730, 1965040180185473309749788, 116359823755204505172646204
Offset: 0

Views

Author

Paul D. Hanna, Feb 27 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 260*x^3 + 4890*x^4 + 110124*x^5 + 2844772*x^6 + 82196424*x^7 + 2613699450*x^8 + ... + a(n)*x^n + ...
where
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^6/dx^6 x^12*A(x)^12)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^(2*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)^2), which begins
B(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 + ... + A360977(n)*x^n + ...
then A(x) = B'(x) and
B(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^5/dx^5 x^11*A(x)^12)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^(2*n))/n! + ... ).

Links

Crossrefs

Cf. A360950, A360973, A360975, A360977.
Cf. A229619, A356848, A088714, A303063.

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^(2*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^2 +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)^(2*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)^2).
(3) B(x - x^2*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^(2*n) / n! ) is the g.f. of A360977.
(4) a(n) = (n+1) * A360977(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1/2)^n, where alfa = 2.498459235192... and c = 0.0920029178453... - Vaclav Kotesovec, Feb 28 2023
alfa = 5*LambertW(1/2) + 1/(1 + LambertW(1/2)). - 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!.

A360978 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)^3).

Original entry on oeis.org

1, 1, 8, 119, 2476, 64370, 1974468, 69109563, 2702001936, 116298977966, 5453395749960, 276403464191890, 15049886389916756, 875933263547340216, 54268470230312961400, 3566244291096016078419, 247800396100716098128236, 18155541676448293842945990
Offset: 1

Views

Author

Paul D. Hanna, Feb 27 2023

Keywords

Comments

a(n) = A360975(n-1)/n for n >= 1.

Examples

			G.f.: A(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 + ...
By definition, A(x - x^2*A'(x)^3) = x, where
A'(x) = 1 + 2*x + 24*x^2 + 476*x^3 + 12380*x^4 + 386220*x^5 + 13821276*x^6 + 552876504*x^7 + ... + A360975(n)*x^n + ...
Also,
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^n/dx^n x^(2*n)*A'(x)^(3*n))/n! + ...
Further,
A(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^(n-1)/dx^(n-1) x^(2*n-1)*A'(x)^(3*n))/n! + ... ).

Links

Crossrefs

Cf. A360975, A229619, A360976, A360977.

Programs

  • PARI
    {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^2*A'^3 +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)^3).
(2) A(x) = x + A(x)^2 * A’(A(x))^3.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A'(x)^(3*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A'(x)^(3*n) / n!.

A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.

Original entry on oeis.org

1, 3, 45, 1113, 36459, 1448568, 66726309, 3469988835, 200242815669, 12670449226269, 871389659249424, 64693985439491127, 5156607707368927875, 439261264283443326927, 39831856169938193953827, 3831650468281643037364389, 389807188331526942149375433
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2023

Keywords

Examples

			G.f.: A(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + a(n)*x^(2*n) + ...
where
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! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)^2), which begins
B(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + 11778989157*x^17 + ... + A361047(n)*x^(2*n-1) + ...
then A(x) = B'(x) and
B(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. A361047, A360950, A360973, A360974, A360975, A360976.
Cf. A229619, A356848, A088714, A303063.

Programs

  • Mathematica
    nt = 40; (* number of terms to produce *)
A[_] = 0;
Do[A[x_] = D[InverseSeries[x - x^3*A[x]^2 + O[x]^k] // Normal, x], {k, 1, 2*nt}];
CoefficientList[A[x^(1/2)], x] (* Jean-François Alcover, Mar 04 2023 *)
  • 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^(3*m)*A^(2*m)/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A^2 +O(x^(2*n+3))))); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^2).
(3) B(x - x^3*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(2*n) / n! ) is the g.f. of A361047.
(4) a(n) = (2*n+1) * A361047(n+1) for n >= 0.
a(n) == 0 (mod 3) for n > 0.
a(n) ~ c * 2^n * n! * n^alfa / LambertW(1/2)^n, where alfa = 1.623844426394406... and c = 0.18597481905555548924712403113114... - Vaclav Kotesovec, Mar 04 2023
alfa = (15*LambertW(1/2) - 1 + 3/(1 + LambertW(1/2)))/4. - Vaclav Kotesovec, Mar 15 2023

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.
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