A045501 Third-from-right diagonal of triangle A121207.

1, 1, 4, 14, 54, 233, 1101, 5625, 30846, 180474, 1120666, 7352471, 50772653, 367819093, 2787354668, 22039186530, 181408823710, 1551307538185, 13756835638385, 126298933271289, 1198630386463990, 11742905240821910

Offset: 1

Author

Henry Gould

Keywords

• easy
• nonn

Comments

With leading 0 and offset 2: number of permutations beginning with 321 and avoiding 1-23. - Ralf Stephan, Apr 25 2004

Second diagonal in table of binomial recurrence coefficients. Related to A040027. - Vladeta Jovovic, Feb 05 2008

Equals eigensequence of triangle A104712. - Gary W. Adamson, Apr 10 2009

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

Links

• 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. A045499, A045500.

Cf. A104712. - Gary W. Adamson, Apr 10 2009

Column k=2 of A124496.

Programs

Mathematica

a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n, k+1]*a[k], {k, 0, n-1}];
Array[a, 22] (* Jean-François Alcover, Jul 14 2018, after Vladeta Jovovic *)

PARI

{a(n)=local(A=x+x^2); for(i=1, n, A=x+x*subst(A, x, x/(1-x+x*O(x^n)))/(1-x)^2); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */

Python

# The function Gould_diag is defined in A121207.
A045501_list = lambda size: Gould_diag(3, size)
print(A045501_list(24)) # Peter Luschny, Apr 24 2016

Formula

a(n+1) = Sum_{k=0..n} binomial(n+2, k+2)*a(k). - Vladeta Jovovic, Nov 10 2003

With offset 2, e.g.f.: x^2 + exp(exp(x))/2 * Integral_{0..x} t^2*exp(-exp(t)+t) dt. - Ralf Stephan, Apr 25 2004

G.f.: A(x) = Sum_{k>=0} x^(k+1)/((1-k*x)^2 * Product_{m=0..k} (1 - m*x)). - Vladeta Jovovic, Feb 05 2008

O.g.f. satisfies: A(x) = x + x*A( x/(1-x) ) / (1-x)^2. - 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