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A271715 Number of set partitions of [3n] with minimal block length multiplicity equal to n.

%I A271715 #11 May 11 2024 18:24:20
%S A271715 1,4,55,1540,67375,4239235,383563180,51925673800,10652498631775,
%T A271715 3139051466175625,1228555090548911125,602267334323068414000,
%U A271715 357161594247065690582500,250870551734754490461422500,205672479804595549379158525000,194557626586812183102927448930000
%N A271715 Number of set partitions of [3n] with minimal block length multiplicity equal to n.
%H A271715 Alois P. Heinz, <a href="/A271715/b271715.txt">Table of n, a(n) for n = 0..200</a>
%F A271715 a(n) = A271424(3n,n).
%F A271715 Recursion: see Maple program.
%F A271715 For n>0, a(n) = (3^n + n!)*(3*n)! / (6^n * (n!)^2). - _Vaclav Kotesovec_, Apr 16 2016
%F A271715 a(n) = A372722(3n,n). - _Alois P. Heinz_, May 11 2024
%p A271715 a:= proc(n) option remember; `if`(n<5,
%p A271715       [1, 4, 55, 1540, 67375][n+1], ((2*(3*n-2))*
%p A271715        (3*n-1)*(n^2-n-9)*a(n-1) -(3*(n-3))*(3*n-1)*
%p A271715        (3*n-4)*(3*n-2)*(3*n-5)*a(n-2))/(4*n*(n-4)))
%p A271715     end:
%p A271715 seq(a(n), n=0..20);
%t A271715 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A271715 b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i&, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]] // Union}]]];
%t A271715 a[n_] := If[n==0, 1, b[3n, 3n, n] - b[3n, 3n, n+1]];
%t A271715 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ in A271424 *)
%Y A271715 Cf. A271424, A372722.
%K A271715 nonn
%O A271715 0,2
%A A271715 _Alois P. Heinz_, Apr 12 2016