A129824 a(n) = Product_{k=0..n} (1 + binomial(n,k)).

2, 4, 12, 64, 700, 17424, 1053696, 160579584, 62856336636, 63812936890000, 168895157342195152, 1169048914836855865344, 21209591746609937928524800, 1010490883477487017627972550656, 126641164340871500483202065902080000, 41817338589698457759723104703370865147904

Offset: 0

Author

Henry Gould, Jun 03 2007

Keywords

• easy
• nonn

Comments

A product analog of the binomial expansion.

The sequence is a special case of a(n) = Product_{k=0..n} (1 + binomial(n,k)*x^k).

Let C be a collection of subsets of an n-element set S. Then a(n) is the number of possible shapes K = (k_0, ..., k_n) of C, where k_i is the number of i-element subsets of S in C. - Gabriel Cunningham (oeis(AT)gabrielcunningham.com), Nov 08 2007

Examples

			a(4) = (1+1)(1+4)(1+6)(1+4)(1+1) = 2*5*7*5*2 = 700.
		

References

• H. W. Gould, A product analog of the binomial expansion, unpublished manuscript, Jun 03 2007.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..69

Crossrefs

Cf. A001142, A055612.

Programs

Magma

A129824:= func< n | (&*[1 + Binomial(n,k): k in [0..n]]) >;
[A129824(n): n in [0..20]]; // G. C. Greubel, Apr 26 2024

Mathematica

Table[Product[1 + Binomial[n,k], {k,0,n}], {n,0,15}] (* Vaclav Kotesovec, Oct 27 2017 *)

PARI

{ a(n) = prod(k=0,n, 1 + binomial(n,k))}
for(n=0,15,print1(a(n),", ")) \\ Paul D. Hanna, Oct 27 2017

SageMath

def A129824(n): return product(1 + binomial(n,k) for k in range(n+1))
[A129824(n) for n in range(21)] # G. C. Greubel, Apr 26 2024

Formula

a(n) = 2*A055612(n). - Reinhard Zumkeller, Jan 31 2015

a(n) ~ exp(n^2/2 + n - 1/12) * A^2 / (n^(n/2 + 1/3) * 2^((n-3)/2) * Pi^((n+1)/2)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 27 2017

Extensions

Corrected and extended by Vaclav Kotesovec, Oct 27 2017