|
%I #46 Feb 05 2022 16:22:25
%S 1,1,4,14,54,233,1101,5625,30846,180474,1120666,7352471,50772653,
%T 367819093,2787354668,22039186530,181408823710,1551307538185,
%U 13756835638385,126298933271289,1198630386463990,11742905240821910
%N Third-from-right diagonal of triangle A121207.
%C With leading 0 and offset 2: number of permutations beginning with 321 and avoiding 1-23. - _Ralf Stephan_, Apr 25 2004
%C Second diagonal in table of binomial recurrence coefficients. Related to A040027. - _Vladeta Jovovic_, Feb 05 2008
%C Equals eigensequence of triangle A104712. - _Gary W. Adamson_, Apr 10 2009
%C 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
%H S. Kitaev, <a href="https://www.mat.univie.ac.at/~slc/wpapers/s48kitaev.html">Generalized pattern avoidance with additional restrictions</a>, Sem. Lothar. Combinat. B48e (2003).
%H S. Kitaev and T. Mansour, <a href="https://arxiv.org/abs/math/0205182">Simultaneous avoidance of generalized patterns</a>, arXiv:math/0205182 [math.CO], 2002.
%F a(n+1) = Sum_{k=0..n} binomial(n+2, k+2)*a(k). - _Vladeta Jovovic_, Nov 10 2003
%F 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
%F 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
%F O.g.f. satisfies: A(x) = x + x*A( x/(1-x) ) / (1-x)^2. - _Paul D. Hanna_, Mar 23 2012
%t a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n, k+1]*a[k], {k, 0, n-1}];
%t Array[a, 22] (* _Jean-François Alcover_, Jul 14 2018, after _Vladeta Jovovic_ *)
%o (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 */
%o (Python)
%o # The function Gould_diag is defined in A121207.
%o A045501_list = lambda size: Gould_diag(3, size)
%o print(A045501_list(24)) # _Peter Luschny_, Apr 24 2016
%Y Cf. A045499, A045500.
%Y Cf. A104712. - _Gary W. Adamson_, Apr 10 2009
%Y Column k=2 of A124496.
%K easy,nonn
%O 1,3
%A _Henry Gould_
%E More terms from _Vladeta Jovovic_, Nov 10 2003
%E Entry revised by _N. J. A. Sloane_, Dec 11 2006
| 