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 1-2 of 2 results.

A250307 Number of 2-colored Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

Original entry on oeis.org

1, 3, 10, 37, 152, 690, 3422, 18257, 103144, 608730, 3713524, 23235490, 148281656, 961255200, 6311395814, 41878914665, 280365966232, 1891270498050, 12842102343820, 87703053156406, 601999871121200, 4150859861430252, 28736613316786220, 199671324115916570
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 20 2015

Keywords

Examples

			a(2) = 10 because we have H1H1, H1H2, H2H1, H2H2, UDH1, UDH2, H1UD, H2UD, UDUD and UUDD.

Crossrefs

Cf. A007564, A224071.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - 2 x - x (1 + 2 x - Sqrt[1 - 8 x + 4 x^2])/(6 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 21 2015 *)
  • Maxima
    a(n):=sum((k+1)*(sum(2^j*(-1)^(-k+j)*binomial(k-j,j),j,0,k))*sum(binomial(j,-n-k+2*j-2)*4^(-n-k+2*j-2)*3^(n-j+1)*binomial(n+1,j),j,0,n+1),k,0,n)/(n+1); /* Vladimir Kruchinin, Mar 08 2016 */

Formula

G.f.: 6/(5-14*x+sqrt(1-8*x+4*x^2))=1/(1-2*x-x*F(x)), where F(x) is the g.f. of the sequence A007564.
a(n) ~ sqrt(1275+746*sqrt(3)) * (2*(2+sqrt(3)))^n / (121*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
a(n) = Sum_{k=0..n}((k+1)*(Sum_{j=0..k}(2^j*(-1)^(-k+j)*binomial(k-j,j)))*Sum_{j=0..n+1}(binomial(j,-n-k+2*j-2)*4^(-n-k+2*j-2)*3^(n-j+1)*binomial(n+1,j)))/(n+1). - Vladimir Kruchinin, Mar 08 2016
Conjecture: 2*n*a(n) +3*(-9*n+8)*a(n-1) +4*(28*n-39)*a(n-2) +4*(-43*n+81)*a(n-3) +64*(n-3)*a(n-4)=0. - R. J. Mathar, Sep 24 2016

Extensions

More terms from Vincenzo Librandi, Apr 21 2015

A257072 Number of 3-colored Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

Original entry on oeis.org

1, 4, 17, 77, 374, 1959, 11085, 67500, 438485, 3004985, 21485222, 158744467, 1202966761, 9297312916, 72981656937, 580105886517, 4658713796790, 37736326098735, 307913254091925, 2528335636842300, 20875157745756429
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 20 2015

Keywords

Examples

			a(1) = 4  because we have H1, H2, H2, UD.

Links

Crossrefs

Cf. A224071, A250307, A059231

Programs

  • Mathematica
    Table[8 DifferenceRoot[Function[{y, n}, {3645 (1 + n) (2 + n) y[n] + (-20736 - 21384 n - 5508 n^2) y[1 + n] + (6120 + 7857 n + 2061 n^2) y[2 + n] + (1330 - 165 n - 175 n^2) y[3 + n] + (-664 - 270 n - 26 n^2) y[4 + n] + 3 (4 + n) (5 + n) y[5 + n] == 0, y[0] == 1/8, y[1] == 1/2, y[2] == 17/8, y[3] == 77/8, y[4] == 187/4}]][k], {k, 0, 20}] (* Benedict W. J. Irwin, May 29 2016 *)
CoefficientList[Series[8/(7 -27*x +Sqrt[1 -10*x +9*x^2]), {x,0,50}], x] (* G. C. Greubel, May 29 2016 *)
  • Maxima
    a(n):=(sum(m*sum(((sum(binomial(j+m,k)*binomial(j-1,j-k)*4^(j-k),k,0,j))*3^(n-j-m)*binomial(n-j,m))/(j+m),j,0,n-m),m,1,n))+3^n; /* Vladimir Kruchinin, Mar 13 2016 */

Formula

G.f.: 8/(7-27*z+sqrt(1-10*z+9*z^2))=1/(1-3*z-z*F(z)), where F(z) is the g.f. of the sequence A059231.
a(n) = (3^n+Sum_{m=1..n}(m*Sum_{j=0..n-m}(((Sum_{k=0..j}(binomial(j+m,k)*binomial(j-1,j-k)*4^(j-k)))*3^(n-j-m)*binomial(n-j,m))/(j+m)))). - Vladimir Kruchinin, Mar 13 2016
a(n) ~ sqrt(2)*3^(2*n-1) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 13 2016
From Benedict W. J. Irwin, May 29 2016: (Start)
Let y(0)=1/8, y(1)=1/2, y(2)=17/8, y(3)=77/8, y(4)=187/4,
Let 3645*(n+1)*(n+2)*y(n)-(5508n^2+21384n+20736)*y(n+1)+(2061n^2+7857n+6120)*y(n+2)-(175n^2+165n-1330)*y(n+3)-(26n^2+270n+664)*y(n+4)+3*(n+4)*(n+5)*y(n+5) = 0,
a(n) = 8*y(n).
(End)
Conjecture: 3*n*a(n) +(-53*n+45)*a(n-1) +2*(151*n-213)*a(n-2) +9*(-73*n+144)*a(n-3) +405*(n-3)*a(n-4)=0. - R. J. Mathar, Sep 24 2016
Showing 1-2 of 2 results.

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