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A272515 Number of set partitions of [n] into three blocks with distinct sizes. 2

%I

%S 60,105,448,2016,4980,15675,61644,155155,482573,1733550,4549808,

%T 13890360,48104628,128949675,392009140,1322692581,3607864403,

%U 10929721440,36245555284,100109572875,302709337515,990788537700,2763564406113,8344789976616,27039048750600

%N Number of set partitions of [n] into three blocks with distinct sizes.

%H Alois P. Heinz, <a href="/A272515/b272515.txt">Table of n, a(n) for n = 6..1000</a>

%F a(n) = n! * [x^n*y^3] Product_{n>=1} (1+y*x^n/n!).

%F Conjecture: a(n) ~ 3^n / 6. - _Vaclav Kotesovec_, Dec 11 2020

%p b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n

%p or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n,i-1,t)+

%p `if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n,i))))

%p end:

%p a:= n-> b(n$2, 3):

%p seq(a(n), n=6..40);

%t b[n_, i_, t_] := b[n, i, t] = If[t > i || t(t+1)/2 > n || t(2i+1-t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]* Binomial[n, i]]]];

%t a[n_] := b[n, n, 3];

%t a /@ Range[6, 40] (* _Jean-Fran├žois Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)

%Y Column k=3 of A131632.

%K nonn

%O 6,1

%A _Alois P. Heinz_, May 01 2016

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Last modified January 17 13:45 EST 2022. Contains 350396 sequences. (Running on oeis4.)