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

0,3

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.

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]

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 *)

Prog

(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

Crossrefs

Cf. A045500, A045501, A346059.

Column k=3 of A124496.

Sequence in context: A152185 A152187 A341920 * A358518 A145932 A026661

Adjacent sequences: A045496 A045497 A045498 * A045500 A045501 A045502

Keyword

easy,nonn

Author

Henry Gould

Extensions

More terms from Vladeta Jovovic, Nov 10 2003

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

Status

approved