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.

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A181136 G.f.: A(x) = Sum_{n>=0} x^n/[Sum_{k=0..n} C(n,k)^3*(-x)^k].

Original entry on oeis.org

1, 1, 2, 10, 92, 1264, 26138, 753322, 28451978, 1385043022, 84971475986, 6393154081582, 580295829204452, 62818032904371952, 8005929383232314294, 1187186361565313907994, 203034917331580351972520
Offset: 0

Views

Author

Paul D. Hanna, Jan 25 2011

Keywords

Comments

Compare the g.f. of this sequence to the identity:
(1-x)/(1-2*x) = Sum_{n>=0} x^n/[Sum_{k=0..n} C(n,k)*(-x)^k].

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1264*x^5 +...
which equals the series:
A(x) = 1 + x/(1-x) + x^2/(1-2^3*x+x^2) + x^3/(1-3^3*x+3^3*x^2-x^3) + x^4/(1-4^3*x+6^3*x^2-4^3*x^3+x^4) + x^5/(1-5^3*x+10^3*x^2-10^3*x^3+5^3*x^4-x^5) +...
		

Crossrefs

Cf. A178324.

Programs

  • Maple
    G:= add(x^n/hypergeom([-n,-n,-n],[1,1],x),n=0..50):
    S:= series(G501,x,51):
    seq(coeff(S,x,n),n=0..50); # Robert Israel, Dec 24 2017
  • PARI
    {a(n)=polcoeff(sum(m=0, n, x^m/sum(k=0, m, binomial(m, k)^3*(-x)^k+x*O(x^n))), n)}

Formula

G.f.: Sum_{n>=0} x^n/hypergeom([-n,-n,-n],[1,1],x). - Robert Israel, Dec 24 2017

A178323 Numbers n such that phi(reversal(n)) + sigma(reversal(n)) = n.

Original entry on oeis.org

572, 592, 5992, 599992, 2014080, 5999992, 594637872, 599999992, 599999999992
Offset: 1

Views

Author

Farideh Firoozbakht, May 28 2010

Keywords

Comments

If n is in the sequence A070272 then reversal(n) is in this sequence. 10 divides all other terms of the sequence. 2014080 is the only known such term.
If p=6*10^n-1 is a prime greater than 5 then reversal(5*p) is in the sequence, see comment lines of A070272.
There is no further term up to 10^9.
10^12 < a(10) <= 1442827967760. - Giovanni Resta, Sep 04 2018

Examples

			2014080 = phi(804102) + sigma(804102), so 2014080 is in the sequence.
		

Crossrefs

Programs

  • Mathematica
    r[n_]:=FromDigits[Reverse[IntegerDigits[n]]];
    Do[If[EulerPhi[r[n]]+DivisorSigma[1,r[n]]==n,Print[n]],{n,1000000000}]

Extensions

a(9) from Giovanni Resta, Sep 04 2018
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