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A362597 Number of parking functions of size n avoiding the patterns 213 and 312.

%I A362597 #16 Jan 11 2024 09:12:56
%S A362597 1,1,3,12,54,259,1293,6634,34716,184389,990711,5372088,29347794,
%T A362597 161317671,891313569,4946324886,27552980088,153982124809,862997075691,
%U A362597 4848839608228,27304369787694,154059320699211,870796075968693,4929937918315522,27950989413184404
%N A362597 Number of parking functions of size n avoiding the patterns 213 and 312.
%H A362597 Ayomikun Adeniran and Lara Pudwell, <a href="https://doi.org/10.54550/ECA2023V3S3R17">Pattern avoidance in parking functions</a>, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
%F A362597 For n>=1, a(n) = Sum_{k=0..n-1} Sum_{i=0..k} binomial(n - 1, i)*(k + 1)*binomial(2*n - 2 - k, n - 1 - k)/n.
%F A362597 D-finite with recurrence (n+1)*a(n) +3*(-4*n+1)*a(n-1) +(34*n-45)*a(n-2) +3*(4*n-17)*a(n-3) +3*(-n+4)*a(n-4)=0. - _R. J. Mathar_, Jan 11 2024
%e A362597 For n=3 the a(3)=12 parking functions, given in block notation, are {1},{2},{3}; {1,2},{},{3}; {1,2},{3},{}; {1},{2,3},{}; {1,2,3},{},{}; {1},{3},{2}; {1,3},{},{2}; {1,3},{2},{}; {2},{3},{1}; {2,3},{},{1}; {2,3},{1},{}; {3},{2},{1}.
%p A362597 A362597 := proc(n)
%p A362597     if n = 0 then
%p A362597         1;
%p A362597     else
%p A362597         add(add(binomial(n - 1, i)*(k + 1)*binomial(2*n - 2 - k, n - 1 - k)/n,i=0..k),k=0..n-1) ;
%p A362597     end if;
%p A362597 end proc:
%p A362597 seq(A362597(n),n=0..60) ; # _R. J. Mathar_, Jan 11 2024
%o A362597 (PARI) a(n)={0^n + sum(k=0, n-1, sum(i=0, k, binomial(n - 1, i)*(k + 1)*binomial(2*n - 2 - k, n - 1 - k)/n))} \\ _Andrew Howroyd_, Apr 27 2023
%Y A362597 Cf. A028365, A362596.
%K A362597 nonn,easy
%O A362597 0,3
%A A362597 _Lara Pudwell_, Apr 27 2023