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 1-3 of 3 results.

A303063 G.f. A(x) satisfies: [x^(n-1)] (1 + x*A(x)^n)^n / A(x)^n = 0 for n>1.

Original entry on oeis.org

1, 1, 3, 17, 151, 1812, 26766, 461302, 8978490, 193200156, 4529641423, 114510000515, 3097375627215, 89116723381943, 2714808312021989, 87242980758842543, 2948618278635037930, 104544558380661516685, 3880035778583841094470, 150451784852703095162304, 6084892588256393044757197, 256294338370540915598727500
Offset: 0

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Author

Paul D. Hanna, Apr 17 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1812*x^5 + 26766*x^6 + 461302*x^7 + 8978490*x^8 + 193200156*x^9 + 4529641423*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in (1 + x*A(x)^n)^n / A(x)^n begins:
n=1: [1, 0, -2, -12, -116, -1475, -22625, -400078, ...];
n=2: [1, 0, -2, -18, -197, -2630, -41347, -742194, ...];
n=3: [1, 0, 0, -15, -228, -3390, -55716, -1022901, ...];
n=4: [1, 0, 4, 0, -178, -3536, -64144, -1228756, ...];
n=5: [1, 0, 10, 30, 0, -2640, -63025, -1327450, ...];
n=6: [1, 0, 18, 78, 369, 0, -45519, -1252758, ...];
n=7: [1, 0, 28, 147, 1008, 5425, 0, -881412, ...];
n=8: [1, 0, 40, 240, 2012, 15080, 91832, 0, ...]; ...
in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1 + x*A(x)^n)^n / A(x)^n = 0 for n>1.

Links

Crossrefs

Cf. A303062, A360578, A360581.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1,n+1, A=concat(A,0); A[m] = Vec( (1 + x*Ser(A)^m)^m/Ser(A)^m )[m]/m ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

a(n) ~ c * n! * n^(3*LambertW(1) + 1/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.03203091421745281863810572012... - Vaclav Kotesovec, Aug 11 2021

A259606 G.f. satisfies: A(x) = Series_Reversion( x - A'(x)*A(x)^2 ).

Original entry on oeis.org

1, 1, 6, 60, 790, 12488, 226176, 4567245, 101057170, 2421311002, 62292579316, 1709994461396, 49844902545256, 1536870296603860, 49965056185462360, 1708221871912841430, 61272046476315041664, 2301058164207089144028, 90309756129843950212480, 3697832634432220792202296
Offset: 1

Views

Author

Paul D. Hanna, Jul 02 2015

Keywords

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 60*x^4 + 790*x^5 + 12488*x^6 + 226176*x^7 + 4567245*x^8 + 101057170*x^9 + 2421311002*x^10 + ...
where
A(x - A'(x)*A(x)^2) = x.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 13*x^4 + 132*x^5 + 1736*x^6 + 27276*x^7 + 490408*x^8 + 9831498*x^9 + 216085602*x^10 + ...
A'(x)*A(x)^2 = x^2 + 4*x^3 + 35*x^4 + 434*x^5 + 6664*x^6 + 119072*x^7 + 2392326*x^8 + 52919810*x^9 + 1271042344*x^10 + ...

Links

Crossrefs

Cf. A360578, A375456.

Programs

  • Mathematica
    a[n_] := Module[{A = x}, Do[A = InverseSeries[x - D[A, x] A^2 + x O[x]^n, x], {n}]; SeriesCoefficient[A, {x, 0, n}]];
Array[a, 25] (* Jean-François Alcover, Sep 28 2020, after PARI *)
  • PARI
    {a(n) = local(A=x); for(i=1,n,A=serreverse(x - A^2*A' +x*O(x^n))); polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x, B=x^2); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, (A')^m*A^(2*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x, B=x^2); for(i=1, n, B=intformal(2*A); A = x*exp(sum(m=1, n, Dx(m-1, (A')^m*A^(2*m)/(m!*x))) +O(x^(n+1)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. satisfies:
(1) A(x) = x + A'(A(x)) * A(A(x))^2.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A'(x)^n * A(x)^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A'(x)^n * A(x)^(2*n) / (n!*x) ).
a(n) ~ c * n! * n^(alfa) / LambertW(1)^n, where alfa = 2.750027682144251700567... and c = 0.005275216890926771261... - Vaclav Kotesovec, Aug 25 2017
alfa = 5*LambertW(1) - 2 + 3/(1 + LambertW(1)). - Vaclav Kotesovec, Mar 13 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 1-3 of 3 results.

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