A045499 Fourth-from-right diagonal of triangle A121207.

1, 1, 5, 20, 85, 400, 2046, 11226, 65676, 407787, 2675410, 18475311, 133843405, 1014271763, 8019687099, 66011609670, 564494701167, 5005880952390, 45958055208576, 436161412834300, 4273045478169842, 43160044390231165

Offset: 0

Author

Henry Gould

Keywords

• easy
• nonn

Comments

With leading 0 and offset 3: number of permutations beginning with 4321 and avoiding 1-23. - Ralf Stephan, Apr 25 2004

a(n) is the number of set partitions of {1,2,...,n+3} in which the last block has length 3; the blocks are arranged in order of their least element. - Don Knuth, Jun 12 2017

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..573
• S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
• S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.

Crossrefs

Cf. A045500, A045501, A346059.

Column k=3 of A124496.

Programs

Maple

A045499 := proc(n)
    option remember ;
    if n =0  then
        1 ;
    else
        add( binomial(n+2,k+3)*procname(k),k=0..n-1) ;
    end if;
end proc: # R. J. Mathar, Jun 03 2014

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[a[k]*Binomial[n+2, k+3], {k, 0, n-1}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 20 2017 *)

PARI

{a(n)=local(A=1+x); for(i=1, n, A=1+x*subst(A, x, x/(1-x+x*O(x^n)))/(1-x)^4); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */

Python

# The function Gould_diag is defined in A121207.
A045499_list = lambda size: Gould_diag(4, size)
print(A045499_list(24)) # Peter Luschny, Apr 24 2016

Formula

a(n+1) = Sum_{k=0..n} binomial(n+3, k+3)*a(k). - Vladeta Jovovic, Nov 10 2003

With offset 3, e.g.f.: x^3 + exp(exp(x))/6 * int[0..x, t^3*exp(-exp(t)+t) dt]. - Ralf Stephan, Apr 25 2004

O.g.f. satisfies: A(x) = 1 + x*A( x/(1-x) ) / (1-x)^4. [Paul D. Hanna, Mar 23 2012]

Extensions

More terms from Vladeta Jovovic, Nov 10 2003

Entry revised by N. J. A. Sloane, Dec 11 2006