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.

A194787 E.g.f. A(x) = G(x)^G(x) where G(x) = 1 + x*G(x)^G(x) is the e.g.f. of A194786.

Original entry on oeis.org

1, 1, 4, 27, 272, 3630, 60714, 1221864, 28779120, 776965680, 23662145160, 802640076480, 30014281406856, 1226796674341056, 54417859649294400, 2603641529587553280, 133660822187138916480, 7328549084322230968320, 427437378614564995967424
Offset: 0

Views

Author

Paul D. Hanna, Sep 02 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 272*x^4/4! + 3630*x^5/5! +...
where A(x) = G(x)^G(x) and G(x) is the e.g.f. of A194786:
G(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1360*x^5/5! +...

Crossrefs

Cf. A194786.

Programs

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

Formula

a(n) = A194786(n+1)/(n+1).

A176118 The n-th derivative of 1/x^x, evaluated at x=1.

Original entry on oeis.org

1, -1, 0, 3, -8, 10, 6, -42, -160, 2952, -27720, 253440, -2553528, 28562664, -349272000, 4618376280, -65615072640, 996952226880, -16133983959744, 277093189849536, -5033937521116800, 96451913892983040, -1943937259314019200, 41112770486238380160
Offset: 0

Views

Author

Jacob Parr (jacobparr1(AT)gmail.com), Apr 09 2010

Keywords

Examples

			E.g.f.: A(x) = 1 - x + 3*x^3/3! - 8*x^4/4! + 10*x^5/5! + 6*x^6/6! - 42*x^7/7! - 160*x^8/8! + 2952*x^9/9! - 27720*x^10/10! + 253440*x^11/11! + ...
The e.g.f. as a power series with reduced fractional coefficients begins
A(x) = 1 - x + 1/2x^3 - 1/3x^4 + 1/12x^5 + 1/120x^6 - 1/120x^7 - 1/252x^8 + 41/5040x^9 - 11/1440x^10 + 2/315x^11 - 106397/19958400x^12 + ...

Links

Crossrefs

Cf. A005727, A008296, A194786.

Programs

  • Maple
    1, seq(simplify(subs(x = 1, diff(x^(-x), `$`(x, n)))), n = 1 .. 22); # Emeric Deutsch, Apr 14 2010
a:= n-> n! *coeftayl(x^(-x), x=1, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 18 2012
  • Mathematica
    NestList[Factor[D[#1, x]] &, 1/x^x, 22] /. (x -> 1) (* Robert G. Wilson v, Feb 03 2013 *)

Formula

E.g.f.: 1 + x*(Q(0) - 1)/(x+1) where Q(k) = 1 - (1+x/(k+1))/(1 - x/(x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 05 2013
a(n) ~ (-1)^(n+1) * n! / n^2. - Vaclav Kotesovec, Sep 03 2014
E.g.f.: 1/(x+1)^(x+1). - Alois P. Heinz, Sep 27 2016
a(n) = Sum_{k=0..n} (-1)^k * A008296(n,k). - Alois P. Heinz, Aug 25 2021
E.g.f.: Sum_{n>=0} (-1)^n * x^n/n! * Product_{k=1..n} (k + x). - Paul D. Hanna, Nov 13 2023

Extensions

Definition edited by Emeric Deutsch, Apr 14 2010
More terms from Emeric Deutsch and R. J. Mathar, Apr 14 2010

A366228 Expansion of e.g.f. A(x) satisfying A(x) = 1 + Integral A(x)^A(x) dx.

Original entry on oeis.org

1, 1, 1, 3, 12, 68, 473, 3998, 39327, 443599, 5629807, 79486044, 1235018598, 20946691457, 385025599130, 7624623236381, 161823815625933, 3664505951884255, 88189911547566082, 2247691180645108608, 60480432646998315279, 1713328345952593367876, 50970518521542636421145
Offset: 0

Views

Author

Paul D. Hanna, Nov 13 2023

Keywords

Comments

(a(n)/(n-1)!)^(1/n) tends to 1.42011... - Vaclav Kotesovec, Nov 15 2023

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 68*x^5/5! + 473*x^6/6! + 3998*x^7/7! + 39327*x^8/8! + 443599*x^9/9! + 5629807*x^10/10! + ...
where A(x) = 1 + Integral A(x)^A(x) dx.
RELATED SERIES.
A(x)^A(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 68*x^4/4! + 473*x^5/5! + 3998*x^6/6! + 39327*x^7/7! + 443599*x^8/8! + ...
log(A(x)) = x + 2*x^3/3! + 3*x^4/4! + 32*x^5/5! + 155*x^6/6! + 1575*x^7/7! + 13573*x^8/8! + 160756*x^9/9! + 1938288*x^10/10! + ...
A(x)^(A(x) - 1) = 1 + 2*x^2/2! + 3*x^3/3! + 32*x^4/4! + 155*x^5/5! + 1575*x^6/6! + 13573*x^7/7! + ...

Links

Crossrefs

Cf. A194786, A176118.

Programs

  • PARI
    {a(n) = my(A=1); for(i=0, n, A = 1 + intformal( A^A +x*O(x^n) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); for(i=0, n, A = exp( intformal( A^(A-1) +x*O(x^n) ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + Integral A(x)^A(x) dx.
(2) A(x) = exp( Integral A(x)^(A(x) - 1) dx ).
(3) A(x) = 1 + Series_Reversion( Integral 1/(1+x)^(1+x) dx ), where 1/(1+x)^(1+x) is the e.g.f. of A176118.
(4) A(x)^A(x) = 1/Sum_{n>=0} (1 - A(x))^n/n! * Product_{k=1..n} (k + A(x)-1) = A'(x). - Paul D. Hanna, Jul 25 2025
Showing 1-3 of 3 results.

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