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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.

A002057 Fourth convolution of Catalan numbers: a(n) = 4*binomial(2*n+3,n)/(n+4).

Original entry on oeis.org

1, 4, 14, 48, 165, 572, 2002, 7072, 25194, 90440, 326876, 1188640, 4345965, 15967980, 58929450, 218349120, 811985790, 3029594040, 11338026180, 42550029600, 160094486370, 603784920024, 2282138106804, 8643460269248, 32798844771700, 124680849918352
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

a(n) is sum of the (flattened) list obtained by the iteration of: replace each integer k with the list 0,...,k+1 on the starting value 0. Length of this list is Catalan(n) or A000108. - Wouter Meeussen, Nov 11 2001
a(n-2) is the number of n-th generation vertices in the tree of sequences with unit increase labeled by 3 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+2,n-1). - Emeric Deutsch, May 30 2004
a(n) = CatalanNumber(n+3) - 2*CatalanNumber(n+2). Proof. From its definition as a convolution of Catalan numbers, a(n) counts lists of 4 Dyck paths of total size (semilength) = n. Connect the 4 paths by 3 upsteps (U) and append 3 downsteps (D). This is a reversible procedure. So a(n) is also the number of Dyck (n+3)-paths that end DDD (D for downstep). Let C(n) denote CatalanNumber(n) (A000108). Since C(n+3) is the total number of Dyck (n+3)-paths and C(n+2) is the number that end UD, we have (*) C(n+3) - C(n+2) is the number of Dyck (n+3)-paths that end DD. Also, (**) C(n+2) is the number of Dyck (n+3)-paths that end UDD (change the last D in a Dyck (n+2)-path to UDD). Subtracting (**) from (*) yields a(n) = C(n+3) - 2C(n+2) as claimed. - David Callan, Nov 21 2006
Convolution square of the Catalan sequence without one of the initial "1"'s: (1 + 4x + 14x^2 + 48x^3 + ...) = (1/x^2) * square(x + 2x^2 + 5x^3 + 14x^4 + ...)
a(n) is the number of binary trees with n+3 internal nodes in which both subtrees of the root are nonempty. Cf. A068875 [Sedgewick and Flajolet]. - Geoffrey Critzer, Jan 05 2013
With offset 4, a(n) is the number of permutations on {1,2,...,n} that are 123-avoiding, i.e., do not contain a three-term monotone subsequence, for which the first ascent is at positions (4,5); for example, there are 48 123-avoiding permutations on n=7 for which the first ascent is at spots (4,5). See Connolly link. There it is shown in general that the k-th Catalan Convolution is the number of 123-avoiding permutations for which the first ascent is at (k, k+1). (For n=k, the first ascent is defined to be at positions (k,k+1) if the permutation is the decreasing permutation with no ascents.) - Anant Godbole, Jan 17 2014
With offset 4, a(n) is the number of permutations on {1,2,...,n} that are 123-avoiding and for which the integer n is in the 4th spot; see Connolly link. - Anant Godbole, Jan 17 2014
a(n) is the number of North-East lattice paths from (0,0) to (n+2,n+2) that have exactly one east step below the subdiagonal y = x-1. Details can be found in Section 3.1 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
a(n) is the number of North-East lattice paths from (0,0) to (n+2,n+2) that bounce off the diagonal y = x to the right exactly once but do not bounce off y = x to the left. Details can be found in Section 4.2 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
a(n) is the number of North-East lattice paths from (0,0) to (n+2,n+2) that horizontally cross the diagonal y = x exactly once but do not cross the diagonal vertically. Details can be found in Section 4.3 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
Apparently also Young tableaux of (non-partition) shape [n+1, 1, 1, n+1], see example file. - Joerg Arndt, Dec 30 2023

Examples

			From _Peter Bala_, Apr 14 2017: (Start)
This sequence appears on the main diagonal of a generalized Catalan triangle. Construct a lower triangular array (T(n,k)), n,k >= 0 by placing the sequence [0,0,0,1,1,1,1,...] in the first column and then filling in the remaining entries in the array using the rule T(n,k) = T(n,k-1) + T(n-1,k). The resulting array begins
  n\k| 0 1  2  3  4   5   6   7  ...
  ---+-------------------------------
   0 | 0
   1 | 0 0
   2 | 0 0  0
   3 | 1 1  1  1
   4 | 1 2  3  4  4
   5 | 1 3  6 10 14  14
   6 | 1 4 10 20 34  48  48
   7 | 1 5 15 35 69 117 165 165
   ...
(see Tedford 2011; this is essentially the array C_4(n,k) in the notation of Lee and Oh). Compare with A279004. (End)

References

  • Pierre de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 11, coefficients of P_4(z).
  • C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146.
  • Robert Sedgewick and Phillipe Flajolet, Analysis of Algorithms, Addison Wesley, 1996, page 225.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

T(n, n+4) for n=0, 1, 2, ..., array T as in A047072. Also a diagonal of A059365 and of A009766.
Cf. A001003.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A000344, A003517, A000588, A003518, A003519, A001392, A001622.
Cf. A145596 (row sums), A279004.

Programs

  • GAP
    List([0..25],n->4*Binomial(2*n+3,n)/(n+4)); # Muniru A Asiru, Mar 05 2018
  • Magma
    [4*Binomial(2*n+3,n)/(n+4): n in [0..30]]; // Vincenzo Librandi, Feb 04 2016
  • Maple
    a := n -> 32*4^n*GAMMA(5/2+n)*(1+n)/(sqrt(Pi)*GAMMA(5+n)):
seq(a(n),n=0..23); # Peter Luschny, Dec 14 2015
A002057List := proc(m) local A, P, n; A := [1]; P := [1,1,1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A002057List(27); # Peter Luschny, Mar 26 2022
  • Mathematica
    Table[Plus@@Flatten[Nest[ #/.a_Integer:> Range[0, a+1]&, {0}, n]], {n, 0, 10}]
Table[4 Binomial[2n+3,n]/(n+4),{n,0,30}] (* or *) CoefficientList[ Series[ (1-Sqrt[1-4 x]+2 x (-2+Sqrt[1-4 x]+x))/(2 x^4),{x,0,30}],x] (* Harvey P. Dale, May 05 2011 *)
  • PARI
    {a(n) = if( n<0, 0, n+=2; 2*binomial(2*n, n-2) / n)}; /* Michael Somos, Jul 31 2005 */
  • PARI
    x='x+O('x^100); Vec((1-(1-4*x)^(1/2)+2*x*(-2+(1-4*x)^(1/2)+x))/(2*x^4)) \\ Altug Alkan, Dec 14 2015
  • SageMath
    [2*(n+1)*catalan_number(n+2)/(n+4) for n in (0..30)] # G. C. Greubel, May 27 2022

Formula

a(n) = A033184(n+4, 4) = 4*binomial(2*n+3, n)/(n+4) = 2*(n+1)*A000108(n+2)/(n+4).
G.f.: c(x)^4 with c(x) g.f. of A000108 (Catalan).
Row sums of A145596. Column 4 of A033184. By specializing the identities for the row polynomials given in A145596 we obtain the results a(n) = Sum_{k = 0..n} (-1)^k*binomial(n+1,k+1)*a(k)*4^(n-k) and a(n) = Sum_{k = 0..floor(n/2)} binomial(n+1,2*k+1) * Catalan(k+1) * 2^(n-2*k). From the latter identity we can derive the congruences a(2n+1) == 0 (mod 4) and a(2n) == Catalan(n+1) (mod 4). It follows that a(n) is odd if and only if n = (2^m - 4) for some m >= 2. - Peter Bala, Oct 14 2008
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=3, a(n-3) = (-1)^(n-3) * coeff(charpoly(A,x), x^3). - Milan Janjic, Jul 08 2010
G.f.: (1-sqrt(1-4*x) + 2*x*(-2+sqrt(1-4*x) + x))/(2*x^4). - Harvey P. Dale, May 05 2011
a(n+1) = A214292(2*n+4,n). - Reinhard Zumkeller, Jul 12 2012
D-finite with recurrence: (n+4)a(n) = 8*(2*n-1)*a(n-3) - 20*(n+1)*a(n-2) + 4*(2*n+5)*a(n-1). - Fung Lam, Jan 29 2014
D-finite with recurrence: (n+4)*a(n) - 2*(3*n+7)*a(n-1) + 4*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Jun 03 2014
Asymptotics: a(n) ~ 4^(n+3)/sqrt(4*Pi*n^3). - Fung Lam, Mar 31 2014
a(n) = 32*4^n*Gamma(5/2+n)*(1+n)/(sqrt(Pi)*Gamma(5+n)). - Peter Luschny, Dec 14 2015
a(n) = C(n+1) - 2*C(n) where C is Catalan number A000108. Yuchun Ji, Oct 18 2017 [Note: Offset is off by 2]
E.g.f.: d/dx ( 2*exp(2*x)*BesselI(2,2*x)/x ). - Ilya Gutkovskiy, Nov 01 2017
From Bradley Klee, Mar 05 2018: (Start)
With F(x) = 16/(1+sqrt(1-4*x))^4 g.f. of A002057, xi(x) = F(x/4)*(x/4)^2, K(16*x) = 2F1(1/2,1/2;1;16*x) g.f. of A002894, q(x) g.f. of A005797, and q'(x) g.f. of A274344:
K(x) = (1+sqrt(xi(x)))*K(xi(x)).
2*K(1-x) = (1+sqrt(xi(x)))*K(1-xi(x)).
q(x) = sqrt(q(xi(16*x)/16)) = q'(xi(16*x)/16)/sqrt(xi(16*x)/16). (End)
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=0} 1/a(n) = 5/4 + Pi/(18*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 183*log(phi)/(25*sqrt(5)) - 77/100, where phi is the golden ratio (A001622). (End)
a(n) = Integral_{x=0..4} x^n*W(x) dx where W(x) = -x^(3/2)*(1 - x/2)*sqrt(4 - x)/Pi, defined on the open interval (0,4). - Karol A. Penson, Nov 13 2022

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