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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A050168 a(0) = 1; for n > 0, a(n) = binomial(n, floor(n/2)) + binomial(n-1, floor(n/2)).

Original entry on oeis.org

1, 2, 3, 5, 9, 16, 30, 55, 105, 196, 378, 714, 1386, 2640, 5148, 9867, 19305, 37180, 72930, 140998, 277134, 537472, 1058148, 2057510, 4056234, 7904456, 15600900, 30458900, 60174900, 117675360, 232676280, 455657715, 901620585, 1767883500
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

a(n) = number of symmetric Dyck (n+1)-paths which either start UD or are prime, i.e., do not return to ground level until the terminal point. For example, a(2)=3 counts UUUDDD, UUDUDD, UDUDUD. - David Callan, Dec 09 2004
a(n) = number of symmetric Dyck (n+1)-paths that first return to ground level either right away or not until the very end, i.e., that remain Dyck paths when either the first two steps or the first and last steps are deleted. For example, a(2)=3 counts UUUDDD, UUDUDD, UDUDUD. - David Callan, Mar 02 2005
Hankel transform has g.f. (1-x(1+x)^2)/(1-x^2(1-x^2)). - Paul Barry, Sep 13 2007

Links

Crossrefs

Maximum element in n-th row of A029653 (generalized Pascal triangle).
Cf. A001405.

Programs

  • Haskell
    a050168 n = a050168_list !! n
a050168_list = 1 : zipWith (+) a001405_list (tail a001405_list)
-- Reinhard Zumkeller, Mar 04 2012
  • Magma
    m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1+x)/(2*x)*(Sqrt((1+2*x)/(1-2*x))-1))); // G. C. Greubel, Oct 26 2018
  • Mathematica
    CoefficientList[(1+x)/(2x) (Sqrt[(1+2x)/(1-2x)]-1) + O[x]^34, x] (* Jean-François Alcover, Aug 04 2018, after Sergei N. Gladkovskii *)
  • PARI
    x='x+O('x^40); Vec((1+x)/(2*x)*(sqrt((1+2*x)/(1-2*x))-1)) \\ G. C. Greubel, Oct 26 2018

Formula

Asymptotic to c*2^n/sqrt(n) where c = (3/4)*sqrt(2/Pi) = 0.598413... - Benoit Cloitre, Jan 13 2003
For n > 0: a(n) = A208976(n-1) + 1. -Reinhard Zumkeller, Mar 04 2012
Conjecture: (n+1)*a(n) + (n-3)*a(n-1) + 2*(-2*n+1)*a(n-2) + 4*(-n+3)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
G.f.: (1+x)/(2*x)*(sqrt((1+2*x)/(1-2*x))-1). - Sergei N. Gladkovskii, Jul 26 2013
G.f.: (1+x)/( W(0)*(1-2*x)*x) - (1+x)/(2*x), where W(k)= 1 + 1/(1 - 2*x/(2*x + (k+1)/(x*(2*k+1))/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013

A208101 Triangle read by rows: T(n,0) = 1; for n > 0: T(n,1) = n, for n>1: T(n,n) = T(n-1,n-2); T(n,k) = T(n-2,k-1) + T(n-1,k) for k: 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 1, 4, 3, 5, 2, 1, 5, 4, 9, 5, 5, 1, 6, 5, 14, 9, 14, 5, 1, 7, 6, 20, 14, 28, 14, 14, 1, 8, 7, 27, 20, 48, 28, 42, 14, 1, 9, 8, 35, 27, 75, 48, 90, 42, 42, 1, 10, 9, 44, 35, 110, 75, 165, 90, 132, 42, 1, 11, 10, 54, 44, 154, 110
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 04 2012

Keywords

Comments

Another variant of Pascal's triangle, cf. A007318.

Examples

			The triangle begins:
0:                    1
1:                  1   1
2:                1   2   1
3:              1   3   2   2
4:            1   4   3   5   2
5:          1   5   4   9   5   5
6:        1   6   5  14   9  14   5
7:      1   7   6  20  14  28  14  14
8:    1   8   7  27  20  48  28  42  14
9:  1   9   8  35  27  75  48  90  42  42

Links

Crossrefs

Cf. A208976 (row sums), A101461 (row max), A208983 (central), A208355 (right edge), A074909.

Programs

  • Haskell
    a208101 n k = a208101_tabl !! n !! k
a208101_row n = a208101_tabl !! n
a208101_tabl =  iterate
   (\row -> zipWith (+) ([0,1] ++ init row) (row ++ [0])) [1]
  • Mathematica
    T[, 0] = 1; T[n, 1] := n; T[n_, n_] := T[n-1, n-2]; T[n_, k_] /; 1Jean-François Alcover, Feb 03 2018 *)
Showing 1-2 of 2 results.

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