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 1-5 of 5 results.

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

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

A360579 Expansion of A(x) satisfying A(x) = Series_Reversion( x - x^3 * A'(x)/A(x) ).

Original entry on oeis.org

1, 1, 3, 15, 105, 941, 10227, 130103, 1890785, 30848357, 557693603, 11059808615, 238659220361, 5566711614125, 139564620135715, 3742989867108071, 106932082058345601, 3242189373760912485, 103987607657060861139, 3517689685292365948343, 125173307497940331598857
Offset: 1

Views

Author

Paul D. Hanna, Feb 22 2023

Keywords

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 15*x^4 + 105*x^5 + 941*x^6 + 10227*x^7 + 130103*x^8 + 1890785*x^9 + 30848357*x^10 + ...
Related series.
Let B(x) = x*A'(x)/A(x), then B(x) is the g.f. of A356848,
B(x) = 1 + x + 5*x^2 + 37*x^3 + 353*x^4 + 4061*x^5 + 54221*x^6 + 820205*x^7 + 13829377*x^8 + 256853629*x^9 + ... + A356848(n)*x^n + ...
such that A( x - x^2*B(x) ) = x,
and B(x) is defined by
B(x) = 1 + x*[(d/dx x*B(x)) + (d^2/dx^2 x^3*B(x)^2)/2! + (d^3/dx^3 x^5*B(x)^3)/3! + (d^4/dx^4 x^7*B(x)^4)/4! + (d^5/dx^5 x^9*B(x)^5)/5! + (d^6/dx^6 x^11*B(x)^6)/6! + ... + (d^n/dx^n x^(2*n-1)*B(x)^n)/n! + ...].
Further,
Series_Reversion(A(x)) = x - x^2 - x^3 - 5*x^4 - 37*x^5 - 353*x^6 - 4061*x^7 - 54221*x^8 - 820205*x^9 + ... + -A356848(n)*x^(n+2) + ...
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 64*x^6 + 441*x^7 + 3795*x^8 + 39504*x^9 + 483852*x^10 + ...
A'(A(x)) = 1 + 2*x + 11*x^2 + 84*x^3 + 798*x^4 + 9000*x^5 + 117232*x^6 + 1730560*x^7 + 28543340*x^8 + ...
A(A(x)) = x + 2*x^2 + 8*x^3 + 46*x^4 + 342*x^5 + 3118*x^6 + 33730*x^7 + 423014*x^8 + 6042106*x^9 + ...
A'(A(x))/A(A(x)) = 1/x + 3*x + 32*x^2 + 368*x^3 + 4752*x^4 + 68556*x^5 + 1095192*x^6 + 19216988*x^7 + ...

Links

Crossrefs

Cf. A356848, A360578.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A=serreverse(x - x^3*A'/A +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, x^(3*m)*(A')^m/A^m/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); for(i=1, n, A = x*exp(sum(m=1, n, Dx(m-1, x^(3*m-1)*(A')^m/A^m/m!)) +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 - x^3 * A'(x)/A(x) ) = x.
(2) A(x) = x + A(x)^3 * A'(A(x)) / A(A(x)).
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n) * (A'(x)/A(x))^n / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * (A'(x)/A(x))^n / n! ).
(5) B(x) = 1 + x*Sum_{n>=1} d^n/dx^n x^(2*n-1) * B(x)^n / n!, where B(x) = x*A'(x)/A(x) is the g.f. of A356848.
a(n) ~ c * n! / (n^(2*(1 - LambertW(1))) * LambertW(1)^n), where c = 0.23898347792869028031... - Vaclav Kotesovec, Feb 23 2023
Showing 1-5 of 5 results.

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