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

0,3

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.

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

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 *)

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)^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

Crossrefs

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

Column k=4 of A124496.

Sequence in context: A249792 A002912 A030297 * A360057 A109115 A038176

Adjacent sequences: A045497 A045498 A045499 * A045501 A045502 A045503

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