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.

A126221 a(n)=c(n)+c(n-1)+2*c(n-2)+4*c(n-3)+8*c(n-4)+...+2^(n-2)*c(1)+2^(n-1)*c(0), where c(k) are the Catalan numbers (A000108).

Original entry on oeis.org

1, 2, 5, 13, 35, 98, 286, 869, 2739, 8910, 29754, 101498, 352222, 1239332, 4410204, 15840813, 57344451, 208976022, 765945954, 2821516398, 10439890026, 38781926652, 144580149924, 540737349858, 2028319233390, 7628680720908
Offset: 0

Views

Author

Emeric Deutsch, Dec 28 2006

Keywords

Comments

Row sums of A125177.
Equals the eigensequence of a triangle with A000108 as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010

Examples

			a(4)=35 because c(4)+c(3)+2*c(2)+4*c(1)+8*c(0) = 14+5+2*2+4*1+8*1 = 35.

Crossrefs

Cf. A125177.

Programs

  • Maple
    c:=n->binomial(2*n,n)/(n+1): a:=n->c(n)+sum(2^(n-j-1)*c(j),j=0..n-1): seq(a(n),n=0..30);

Formula

G.f.: (1-x)*(1-sqrt(1-4*x)) / (2*x*(1-2*x)).
D-finite with recurrence (n+1)*a(n) +(-7*n+1)*a(n-1) +2*(7*n-8)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jul 22 2022

A277871 a(n) = Sum_{i=0..n+1} binomial(2*n-i,n-i+1)*Catalan(i).

Original entry on oeis.org

1, 4, 16, 66, 279, 1203, 5275, 23474, 105853, 483108, 2229253, 10390691, 48879588, 231879456, 1108473015, 5335987930, 25849521109, 125945214309, 616833862018, 3035286848660, 14999774773110, 74413424196360, 370463714276625, 1850251796668899
Offset: 0

Views

Author

Vladimir Kruchinin, Nov 02 2016

Keywords

Comments

T(2*n+1,n) is diagonal of triangle A125177.

Links

Crossrefs

Cf. A000108, A125177.

Programs

  • Mathematica
    Table[Sum[Binomial[2*i, i]*Binomial[2*n-i, n-i+1]/(i+1), {i, 0, n+1}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2016 *)
  • Maxima
    a(n):=sum((binomial(2*i,i)*binomial(2*n-i,n-i+1))/(i+1),i,0,n+1);
  • PARI
    x='x+O('x^50); Vec((4*x*(1-sqrt(1-2*(1-sqrt(1-4*x)))))/(1-sqrt(1-4*x))^3/sqrt(1-4*x)-1/x) \\ G. C. Greubel, Apr 09 2017

Formula

G.f.: (4*x*(1-sqrt(1-2*(1-sqrt(1-4*x)))))/(1-sqrt(1-4*x))^3/sqrt(1-4*x)-1/x.
a(n) ~ 2^(4*n+1/2) / (sqrt(Pi) * n^(3/2) * 3^(n-3/2)). - Vaclav Kotesovec, Nov 05 2016

A112413 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and starting with exactly k UD's, where U=(1,1), D=(1,-1) (0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 9, 3, 1, 0, 1, 28, 9, 3, 1, 0, 1, 90, 28, 9, 3, 1, 0, 1, 297, 90, 28, 9, 3, 1, 0, 1, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 3432, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 11934, 3432, 1001, 297, 90, 28, 9, 3, 1, 0, 1, 41990, 11934, 3432, 1001, 297, 90, 28, 9, 3, 1, 0
Offset: 0

Views

Author

Emeric Deutsch, Dec 08 2005

Keywords

Comments

All columns, except for initial terms, yield A000245. Row sums yield the Catalan numbers (A000108).
Riordan array ((1-x)*c(x),x), c(x) the g.f. of A000108; equal to A125177*A130595. - Philippe Deléham, Dec 08 2009

Examples

			T(5,2)=3 because we have UDUDUUDDUD, UDUDUUDUDD and UDUDUUUDDD, where U=(1,1), D=(1,-1).
Triangle begins:
   1;
   0, 1;
   1, 0, 1;
   3, 1, 0, 1;
   9, 3, 1, 0, 1;
  28, 9, 3, 1, 0, 1;

Crossrefs

Cf. A000108, A000245.

Programs

  • Maple
    T:=proc(n,k) local c: c:=n->binomial(2*n,n)/(n+1): if k

Formula

T(n,k) = c(n-k) - c(n-k-1), where c(n) = binomial(2n, n)/(n+1) is the n-th Catalan number. G.f. = (1-z)*C/(1-tz), where C = (1-sqrt(1-4z))/(2z) is the Catalan function.
Showing 1-3 of 3 results.

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