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.

A047749 If n = 2*m then a(n) = binomial(3*m, m)/(2*m+1), if n=2*m+1 then a(n) = binomial(3*m+1, m+1)/(2*m+1).

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 12, 30, 55, 143, 273, 728, 1428, 3876, 7752, 21318, 43263, 120175, 246675, 690690, 1430715, 4032015, 8414640, 23841480, 50067108, 142498692, 300830572, 859515920, 1822766520, 5225264024, 11124755664, 31983672534, 68328754959, 196947587823, 422030545335
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Hankel transform appears to be a signed aerated version of A059492. - Paul Barry, Apr 16 2008
Row sums of inverse Riordan array (1, x*(1-x^2))^(-1). - Paul Barry, Apr 16 2008
a(n) is the number of permutations of length n avoiding 213 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
From David Callan, Aug 22 2014: (Start)
a(n) is the number of ordered trees (A000108) with n vertices in which every non-root non-leaf vertex has exactly one leaf child (no restriction on its non-leaf children). For example, a(4) counts the 3 trees
| |
\/ \|/ \/
(End)
From Emeric Deutsch, Oct 28 2014: (Start)
a(n) is the number of symmetric ternary trees having n internal nodes.
a(n) is the number of symmetric non-crossing rooted trees having n edges.
a(n) is the number of symmetric even trees having 2n edges.
a(n) is the number of symmetric diagonally convex directed polyominoes having n diagonals.
(End)
For the above 4 items see the Deutsch-Feretic-Noy reference.
a(n) is also the number of self-dual labeled non-crossing trees with n edges. See my paper in the links section. - Nikos Apostolakis, Jun 11 2019
Number of achiral polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. An achiral polyomino is identical to its reflection. - Robert A. Russell, Jan 20 2024

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 7*x^5 + 12*x^6 + 30*x^7 + 55*x^8 + ...

Links

Crossrefs

Column k=3 of A369929 and k=4 of A370062.
Cf. A000108, A032766, A059492.
Cf. A006013 is the odd-indexed terms of this sequence.
Polyominoes: A005034 (oriented), A005036 (unoriented), A369315 (chiral), A385149 (asymmetric), A001764 (rooted), A208355(n-1) {3,oo}, A369472 {5,oo}.

Programs

  • Magma
    G:=Gamma; [Round((1+(-1)^n)*G(3*n/2+1)/(G(n/2+1)*Factorial(n+1)) + (1-(-1)^n)*G((3*n+1)/2)/(G((n+3)/2)*Factorial(n)))/2: n in [0..35]]; // G. C. Greubel, Jul 07 2019
  • Maple
    A047749 := proc(m) if m mod 2 = 1 then x := (m-1)/2; RETURN((3*x+1)!/((x+1)!*(2*x+1)!)); fi; x := m/2; RETURN((3*x)!/(x!*(2*x+1)!)); end;
A047749 := proc(m) local x; if m mod 2 = 1 then x := (m-1)/2; RETURN((3*x+1)!/((x+1)!*(2*x+1)!)); fi; RETURN(A001764(m/2)); end;
  • Mathematica
    a[ n_] := If[ n < 1, Boole[n == 0], SeriesCoefficient[ InverseSeries[ Series[ (x + 2 x^2) / (1 + x)^3, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Oct 29 2014 *)
Table[If[OddQ[n],2Binomial[(3n-1)/2,(n-1)/2],Binomial[3n/2,n/2]]/(n+1),{n,0,40}] (* Robert A. Russell, Jan 19 2024 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^2*subst(A,x,-x+x*O(x^n)));polcoeff(A,n)} \\ Paul D. Hanna, Sep 20 2009
  • PARI
    x='x+O('x^66);
C(x)=serreverse(x-x^3); /* =x+x^3+3*x^5+12*x^7+55*x^9 +..., cf. A001764 */
s=1/(1-C(x)); /* g.f. */
Vec(s) /* Joerg Arndt, Apr 16 2011 */
  • PARI
    apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\2, 3, n%2+1); \\ Seiichi Manyama, Jul 20 2025
  • Python
    from math import comb
def A047749(n): return comb(n+(a:=n>>1),a+(b:=n&1))//(n+1-b) # Chai Wah Wu, Jul 30 2022
  • Sage
    def A047749_list(n) :
    D = [0]*n; D[1] = 1
    R = []; b = False; h = 1
    for i in range(n) :
        for k in (1..h) :
            D[k] = D[k] + D[k-1]
        R.append(D[h])
        if b : h += 1
        b = not b
    return R
A047749_list(35) # Peter Luschny, May 03 2012
  • Sage
    [1]+[((1+(-1)^n)*binomial(3*n/2,n/2)/(n+1) + (1-(-1)^n)* binomial((3*n-1)/2, (n+1)/2)/n)/2 for n in (1..35)] # G. C. Greubel, Jul 07 2019

Formula

G.f. is 1+Z, where Z satisfies x*Z^3 + (3*x-2)*Z^2 + (3*x-1)*Z + x = 0. Equivalently, the g.f. Y satisfies x*Y^3 - 2*Y^2 + 3*Y - 1 = 0. - Vladeta Jovovic, Dec 06 2002
Reversion of g.f. (x-2*x^2)/(1-x)^3 (ignoring signs). - Ralf Stephan, Mar 22 2004
G.f.: (4/(3*x))*(sin((1/3)*asin(sqrt(27*x^2/4))))^2 +(2/sqrt(3*x^2))*sin((1/3)*asin(sqrt(27*x^2/4))). - Paul Barry, Nov 08 2006
G.f.: 1/(1-2*sin(asin(3*sqrt(3)*x/2)/3)/sqrt(3)). - Paul Barry, Apr 16 2008
From Paul D. Hanna, Sep 20 2009: (Start)
G.f. satisfies: A(x) = 1 + x*A(x)^2*A(-x);
also, A(x)*A(-x) = B(x^2) where B(x) = 1 + x*B(x)^3 = g.f. of A001764. (End)
G.f.: 1/(1-C(x)) where C(x) = Reverse(x-x^3) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 + ... (cf. A001764). - Joerg Arndt, Apr 16 2011
G.f.: G(z^2)+z*G(z^2)^2, where G(z) = 1 + z*G(z)^3, the generating function for A001764. - Robert A. Russell, Jan 26 2024
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) is the upper left term in M^n, M = the infinite square production matrix:
1, 1, 0, 0, 0, 0, ...
0, 0, 1, 0, 0, 0, ...
1, 1, 0, 1, 0, 0, ...
0, 0, 1, 0, 1, 0, ...
1, 1, 0, 1, 0, 1, ...
... (End)
Conjecture D-finite with recurrence: 8*n*(n+1)*a(n) + 36*n*(n-2)*a(n-1) - 6*(9*n^2-18*n+14)*a(n-2) - 27*(3*n-7)*(3*n-8)*a(n-3) = 0. - R. J. Mathar, Dec 19 2011
0 = a(n)*(+7308954*a(n+2) - 16659999*a(n+3) - 4854519*a(n+4) + 6201838*a(n+5)) + a(n+1)*(-406053*a(n+2) - 1627560*a(n+3) + 1683538*a(n+4) - 245747*a(n+5)) + a(n+2)*(+45117*a(n+2) + 235870*a(n+3) + 173953*a(n+4) - 484295*a(n+5)) + a(n+3)*(-41820*a(n+3) - 50184*a(n+4) + 22304*a(n+5)) for all n in Z if a(-1) = -2/3. - Michael Somos, Oct 29 2014
a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} (-1)^i * a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Jul 28 2021
a(n) = binomial(A032766(n),floor((n+1)/2))/(2*floor(n/2)+1). - Miko Labalan, Nov 28 2023
a(n) = 2*A005036(n) - A005034(n) = A005034(n) - 2*A369315(n) = A005036(n) - A369315(n). - Robert A. Russell, Jan 20 2024
From Robert A. Russell, Mar 20 2024: (Start)
a(n) = U(n) in the Beineke and Pippert link.
G.f.: E(1)(t*E(3)(t^2)) (second entry in Table 1), where E(d)(t) is defined in formula 3 of Hering link. (End)
From Robert A. Russell, Jul 15 2024: (Start)
a(2m) = A001764(m) ~ (3^3/2^2)^m*sqrt(3/(2*Pi*(2*m)^3)).
a(n+2)/a(n) ~ 27/4; a(2m+1)/a(2m) ~ 3; a(2m)/a(2m-1) ~ 9/4. (End)
a(n) ~ 3^((6n+3)/4)/(sqrt(Pi)*2^((2n-1)/2)*(2n+1)^(3/2)). - Miko Labalan, Dec 05 2024
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} a(2*k) * a(n-1-2*k). - Seiichi Manyama, Jul 07 2025

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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