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A272514 Number of set partitions of [n] into two blocks with distinct sizes.

%I A272514 #17 Jul 15 2024 10:22:16
%S A272514 3,4,15,21,63,92,255,385,1023,1585,4095,6475,16383,26332,65535,106761,
%T A272514 262143,431909,1048575,1744435,4194303,7036529,16777215,28354131,
%U A272514 67108863,114159427,268435455,459312151,1073741823,1846943452,4294967295,7423131481,17179869183
%N A272514 Number of set partitions of [n] into two blocks with distinct sizes.
%H A272514 Alois P. Heinz, <a href="/A272514/b272514.txt">Table of n, a(n) for n = 3..1000</a>
%F A272514 a(n) = n! * [x^n*y^2] Product_{n>=1} (1+y*x^n/n!).
%F A272514 a(n) = Sum_{i=1..floor((n-1)/2)} binomial(n,i). - _Wesley Ivan Hurt_, Nov 15 2017
%F A272514 a(n) ~ 2^(n-1). - _Vaclav Kotesovec_, Dec 11 2020
%p A272514 b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
%p A272514       or t*(2*i+1-t)/2<n, 0, `if`(n=0, 1, b(n,i-1,t)+
%p A272514       `if`(i>n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
%p A272514     end:
%p A272514 a:= n-> b(n$2, 2):
%p A272514 seq(a(n), n=3..40);
%t A272514 Table[Sum[Binomial[n, i], {i, Floor[(n - 1)/2]}], {n, 3, 35}] (* _Michael De Vlieger_, Nov 15 2017 *)
%o A272514 (Magma) [(&+[Binomial(n,j): j in [1..Floor((n-1)/2)]]): n in [3..40]]; // _G. C. Greubel_, Jul 14 2024
%o A272514 (SageMath)
%o A272514 def A272514(n): return sum( binomial(n,j) for j in range(1,1+((n-1)//2)))
%o A272514 [A272514(n) for n in range(3,31)] # _G. C. Greubel_, Jul 14 2024
%Y A272514 Column k=2 of A131632.
%K A272514 nonn,easy
%O A272514 3,1
%A A272514 _Alois P. Heinz_, May 01 2016