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.

A144003 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral 1/A(x)^3 dx ).

Original entry on oeis.org

1, 1, 3, 24, 339, 7101, 200961, 7256277, 321662502, 17029233774, 1054682936433, 75199620036177, 6094256204678922, 555527437385512095, 56468189426338157580, 6353824422205136494044, 786458781488123265873519
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2008

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 339*x^4/4! + 7101*x^5/5! + 200961*x^6/6! + 7256277*x^7/7! + 321662502*x^8/8! + ...
where A(x) = 1 + Series_Reversion( Integral 1/A(x)^3 dx ).
RELATED SERIES.
Integral 1/A(x)^3 dx = x - 3*x^2/2! + 3*x^3/3! - 24*x^4/4! - 261*x^5/5! - 6543*x^6/6! - 202671*x^7/7! - 7911351*x^8/8! + ...
where Integral 1/A(x)^3 dx = Series_Reversion(A(x) - 1).
A(A(x) - 1) = 1 + x + 6*x^2/2! + 75*x^3/3! + 1479*x^4/4! + 40617*x^5/5! + 1447785*x^6/6! + 64027656*x^7/7! + 3404869020*x^8/8! + ...
A(A(x) - 1)^3 = 1 + 3*x + 24*x^2/2! + 339*x^3/3! + 7101*x^4/4! + ...
where A(A(x) - 1)^3 = d/dx A(x).

Links

Crossrefs

Cf. A001028, A144002, A144004.

Programs

  • PARI
    {a(n) = my(A=1+x+x*O(x^n)); for(i=0,n, A = 1 + serreverse(intformal(1/A^3))); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: A'(x) = A(A(x) - 1)^3. - Paul D. Hanna, Aug 26 2024

A144004 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral 1/A(x)^4 dx ).

Original entry on oeis.org

1, 1, 4, 44, 856, 24664, 958592, 47463936, 2881313024, 208638075392, 17654019768320, 1717961286944768, 189836122499649536, 23574107397852049408, 3261667682403085852672, 499151625979680748978176
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2008

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 44*x^3/3! + 856*x^4/4! + 24664*x^5/5! + 958592*x^6/6! + 47463936*x^7/7! + 2881313024*x^8/8! + ...

Crossrefs

Cf. A001028, A144002, A144003.

Programs

  • PARI
    {a(n) = my(A=1+x+x*O(x^n)); for(i=0,n, A = 1 + serreverse(intformal(1/A^4))); n!*polcoef(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: A'(x) = A(A(x) - 1)^4. - Paul D. Hanna, Sep 07 2024

A144005 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral A(-x) dx ).

Original entry on oeis.org

1, 1, 1, 2, 7, 33, 201, 1479, 12842, 127952, 1440989, 18070767, 249766088, 3769280801, 61654447712, 1085974748430, 20485430748783, 411839042136379, 8786499316562396, 198174104269740313, 4708919322491690592
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2008

Keywords

Comments

This is essentially the same sequence as A014623. - N. J. A. Sloane, Jul 24 2022
Compare the definition of e.g.f. A(x) to the trivial statement:
if F(x) = 1 + Series_Reversion( Integral F(-x)^2 dx ) then F(x) = 1/(1-x).

Examples

			E.g.f: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 7*x^4/4! + 33*x^5/5! +...
Let I(x) = Series_Reversion(A(x) - 1) = Integral A(-x) dx then
I(x) = x - x^2/2! + x^3/3! - 2*x^4/4! + 7*x^5/5! - 33*x^6/6! +...

Crossrefs

Cf. A014623; A144002.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0,n,A=1+serreverse(intformal(subst(A,x,-x)^1)));n!*polcoeff(A,n)}

Formula

E.g.f. satisfies: A(x) = 1 + Integral 1/A(1 - A(x)) dx. - Paul D. Hanna, Jul 10 2015
Showing 1-3 of 3 results.

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

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