A292613 a(n) = [x^n] 1/(1-x)^n * Product_{k=1..n} 1/(1-x^k).

1, 2, 7, 25, 92, 343, 1292, 4902, 18703, 71677, 275694, 1063636, 4114131, 15948762, 61946290, 241013869, 939125870, 3664299332, 14314777054, 55982787136, 219158088711, 858728875776, 3367576480747, 13216392846128, 51905939548950, 203989227456894, 802164259099114

Offset: 0

Author

Vaclav Kotesovec, Sep 20 2017

Keywords

• nonn

Comments

Number of ways to pick n units in all partitions of 2n - Olivier Gérard, May 07 2020

Examples

			Illustration of comment for n=3, a(3)=25 :
Among the 11 integer partitions of 6, 3 have at least 3 ones.
3,1,1,1  ;  2,1,1,1,1;  1,1,1,1,1,1;
There are respectively 1, 4 and 20 ways to pick 3 of these.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1663 (first 301 terms from Vaclav Kotesovec)

Crossrefs

Cf. A000041, A001700, A014329, A292463.

Cf. A292508, A322211.

Programs

Mathematica

Table[SeriesCoefficient[1/(1-x)^n*Product[1/(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]

Formula

a(n) ~ c * 4^n / sqrt(Pi*n), where c = 1/(2*QPochhammer[1/2, 1/2]) = 1.7313733097275318057689... - Vaclav Kotesovec, Sep 20 2017

a(n) = A292508(n,n+1). - Alois P. Heinz, Jul 16 2021