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A272514
Number of set partitions of [n] into two blocks with distinct sizes.
3
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
3,1
LINKS
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
MAPLE
b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
or t*(2*i+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))))
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 *)
PROG
(Magma) [(&+[Binomial(n, j): j in [1..Floor((n-1)/2)]]): n in [3..40]]; // G. C. Greubel, Jul 14 2024
(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
CROSSREFS
Column k=2 of A131632.
Sequence in context: A369910 A095799 A109926 * A065942 A369082 A036759
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, May 01 2016
STATUS
approved