A045500 Fifth-from-right diagonal of triangle A121207.

1, 1, 6, 27, 125, 635, 3488, 20425, 126817, 831915, 5744784, 41618459, 315388311, 2493721645, 20526285716, 175529425815, 1556577220651, 14290644428279, 135624265589086, 1328702240382589, 13420603191219111, 139592874355534071

Offset: 0

Author

Henry Gould

Keywords

• easy
• nonn

Comments

With leading 0 and offset 4: number of permutations beginning with 54321 and avoiding 1-23. - Ralf Stephan, Apr 25 2004

a(n) is the number of set partitions of {1,2,...,n+4} in which the last block has length 4: the blocks are arranged in order of their least element. - Don Knuth, Jun 12 2017

References

• See also references under sequence A040027.

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], 2014.

Crossrefs

Cf. A040027, A045499, A045501, A121207, A346060.

Column k=4 of A124496.

Programs

Mathematica

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n+3, k+4]*a[k], {k, 0, n-1}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 14 2018, after Vladeta Jovovic *)

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)^5); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */

Python

# The function Gould_diag is defined in A121207.
A045500_list = lambda size: Gould_diag(5, size)
print(A045500_list(24)) # Peter Luschny, Apr 24 2016

Formula

a(n+1) = Sum_{k=0..n} binomial(n+4, k+4)*a(k). - Vladeta Jovovic, Nov 10 2003

With offset 4, e.g.f.: x^4 + exp(exp(x))/24 * int[0..x, t^4*exp(-exp(t)+t) dt]. - Ralf Stephan, Apr 25 2004

O.g.f. satisfies: A(x) = 1 + x*A( x/(1-x) ) / (1-x)^5. - 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