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

3, 4, 15, 21, 63, 92, 255, 385, 1023, 1585, 4095, 6475, 16383, 26332, 65535, 106761, 262143, 431909, 1048575, 1744435, 4194303, 7036529, 16777215, 28354131, 67108863, 114159427, 268435455, 459312151, 1073741823, 1846943452, 4294967295, 7423131481, 17179869183

Offset: 3

Author

Alois P. Heinz, May 01 2016

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=2 of A131632.

Programs

Magma

[(&+[Binomial(n,j): j in [1..Floor((n-1)/2)]]): n in [3..40]]; // G. C. Greubel, Jul 14 2024

Maple

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 2):
seq(a(n), n=3..40);

Mathematica

Table[Sum[Binomial[n, i], {i, Floor[(n - 1)/2]}], {n, 3, 35}] (* Michael De Vlieger, Nov 15 2017 *)

SageMath

def A272514(n): return sum( binomial(n,j) for j in range(1,1+((n-1)//2)))
[A272514(n) for n in range(3,31)] # G. C. Greubel, Jul 14 2024

Formula

a(n) = n! * [x^n*y^2] Product_{n>=1} (1+y*x^n/n!).

a(n) = Sum_{i=1..floor((n-1)/2)} binomial(n,i). - Wesley Ivan Hurt, Nov 15 2017

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Dec 11 2020