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
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Nov 10 2003
Entry revised by N. J. A. Sloane, Dec 11 2006
STATUS
approved