A180272 a(n) = binomial(n, A002024(n+1)-1) where A002024 is "n appears n times".

1, 1, 2, 3, 6, 10, 20, 35, 56, 84, 210, 330, 495, 715, 1001, 3003, 4368, 6188, 8568, 11628, 15504, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 1184040, 1560780, 2035800, 2629575, 3365856, 4272048, 5379616, 6724520, 30260340, 38608020, 48903492

Offset: 0

Author

Paul D. Hanna, Jan 17 2011

Keywords

• nonn

Comments

Number of subsets of [n] in which exactly half of the elements are triangular numbers: a(6) = 20: {}, {1,2}, {1,4}, {1,5}, {2,3}, {2,6}, {3,4}, {3,5}, {4,6}, {5,6}, {1,2,3,4}, {1,2,3,5}, {1,2,4,6}, {1,2,5,6}, {1,3,4,5}, {1,4,5,6}, {2,3,4,6}, {2,3,5,6}, {3,4,5,6}, {1,2,3,4,5,6}. - Alois P. Heinz, Oct 11 2022

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 +...
Terms are shown below in parenthesis as they appear in Pascals triangle:
(1);
1,(1);
1,(2),1;
1,3,(3),1;
1,4,(6),4,1;
1,5,(10),5,1;
1,6,15,(20),15,6,1;
1,7,21,(35),35,21,7,1;
1,8,28,(56),70,56,28,8,1;
1,9,36,(84),126,126,84,36,9,1;
1,10,45,120,(210),252,210,120,45,10,1; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000217, A002024.

Programs

PARI

{a(n)=binomial(n,(sqrtint(8*n+1)-1)\2)}

Python

from math import comb, isqrt
def A180272(n): return comb(n,(isqrt(n+1<<3)+1>>1)-1) # Chai Wah Wu, Oct 17 2022