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.

A162547 Somos-4 variant: if n!=4k+1, then a(n) = (4*a(n-1)*a(n-3) - 4*a(n-2)^2) / a(n-4), otherwise a(n) = 0, with a(-2) = a(-1) = a(0) = 1.

Original entry on oeis.org

1, 0, -4, -16, -64, 0, 4096, 65536, 1048576, 0, -1073741824, -68719476736, -4398046511104, 0, 72057594037927936, 18446744073709551616, 4722366482869645213696, 0, -1237940039285380274899124224, -1267650600228229401496703205376, -1298074214633706907132624082305024, 0
Offset: 0

Views

Author

Paul Barry, Jul 05 2009

Keywords

Comments

Hankel transform of A101499.
Hankel transform of A124431. - Paul Barry, Oct 02 2009

Links

Crossrefs

Cf. A101499, A124431.

Programs

  • Mathematica
    a[ n_] := With[{m = n + 1, k = Quotient[n + 2, 4]}, Boole[Mod[m, 4] != 2] (-1)^k 4^(k (m - 2 k))]; (* Michael Somos, Jun 26 2017 *)
  • PARI
    {a(n) = my(m=n+1, k=(n+2)\4); (m%4!=2) * (-1)^k * 4^(k*(m - 2*k))}; /* Michael Somos, Jul 20 2014 */

Formula

a(n) = a(-2 - n), a(n) * a(n+5) = 4 * a(n+1) * a(n+4) for all n in Z. - Michael Somos, Jul 20 2014
a(4*n + 1) = 0 for all n in Z. - Michael Somos, Jun 26 2017

A102880 A Chebyshev transform of the first kind of the Catalan numbers.

Original entry on oeis.org

1, 1, 0, 2, 8, 22, 64, 198, 624, 1994, 6464, 21210, 70296, 234990, 791424, 2682894, 9147360, 31347730, 107919232, 373055730, 1294372008, 4506163718, 15735793088, 55105084246, 193471595344, 680891484762, 2401575077568, 8487950090954
Offset: 0

Views

Author

Paul Barry, Jan 15 2005

Keywords

Comments

Image of c(x) under the mapping g(x)->((1-x^2)/(1+x^2))g(x/(1+x^2)).

Crossrefs

Cf. A101499, A102879.

Formula

G.f.: ((1-x^2)/(1+x^2))c(x/(1+x^2)), c(x) the g.f. of the Catalan numbers A000108; a(n)=n*sum{k=0..floor(n/2), C(n-k, k)(-1)^k*C(n-2k)/(n-k)}.
Conjecture: (n+1)*(n-3)*a(n) -2*(2*n-1)*(n-3)*a(n-1) +2*(1-4*n+n^2)*a(n-2) -2*(n-1)*(2*n-7)*a(n-3) +(n-1)*(n-5)*a(n-4)=0. - R. J. Mathar, Nov 09 2012

A242566 Expansion of (1-sqrt(1-(2*(1-sqrt(1-4*x^2)))/x))/2.

Original entry on oeis.org

0, 1, 1, 3, 7, 22, 67, 225, 765, 2704, 9710, 35558, 131859, 494892, 1874901, 7162807, 27558511, 106695148, 415346144, 1624780952, 6383671910, 25179642120, 99670897534, 395810459602, 1576464630375, 6295827843098
Offset: 0

Views

Author

Vladimir Kruchinin, Jun 09 2014

Keywords

Comments

The sequence 1, 1, 3, 7, ... with offset 0 is the Riordan transform with the Riordan matrix A053121 (the inverse of the Chebyshev S matrix A049310) of the Catalan sequence A000108. - Wolfdieter Lang, Feb 18 2017

Crossrefs

Cf. A000108, A049310, A053121, A101499.

Programs

  • Mathematica
    CoefficientList[Series[1/2 - Sqrt[(-2 + x + 2*Sqrt[1-4*x^2])/x]/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 15 2014 *)
  • Maxima
    a(n):=sum(binomial(2*n-4*i-2,n-2*i-1)*binomial(n,i),i,0,(n-1)/2)/(n);
  • PARI
    a(n) = if (n, sum(i=0,(n-1)/2, binomial(2*n-4*i-2,n-2*i-1)*binomial(n,i))/n, 0); \\ Michel Marcus, Jun 09 2014

Formula

a(n) = sum(i=0..(n-1)/2, binomial(2*n-4*i-2,n-2*i-1)*binomial(n,i))/n, n>0, a(0)=0.
G.f. A(x) = x*C(x^2)*C(x*C(x^2)), where C(x) is g.f. A000108.
G.f. A(x) satisfies A(x)=x*(1/(1-A(x))+A(x)^2-A(x)^3).
a(n) ~ 17^(n+1/2) / (sqrt(15*Pi) * n^(3/2) * 4^(n+1)). - Vaclav Kotesovec, Jun 15 2014
Conjecture D-finite with recurrence: 2*n*(2*n+1)*a(n) +(-49*n^2+97*n-36)*a(n-1) +12*(10*n^2-42*n+41)*a(n-2) +4*(49*n-97)*(n-3)*a(n-3) -544*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jan 25 2020
Showing 1-3 of 3 results.

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