A129824 - OEIS

%I #21 Apr 26 2024 05:53:54

%S 2,4,12,64,700,17424,1053696,160579584,62856336636,63812936890000,

%T 168895157342195152,1169048914836855865344,21209591746609937928524800,

%U 1010490883477487017627972550656,126641164340871500483202065902080000,41817338589698457759723104703370865147904

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

%C A product analog of the binomial expansion.

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

%C 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

%D H. W. Gould, A product analog of the binomial expansion, unpublished manuscript, Jun 03 2007.

%H G. C. Greubel, <a href="/A129824/b129824.txt">Table of n, a(n) for n = 0..69</a>

%F a(n) = 2*A055612(n). - _Reinhard Zumkeller_, Jan 31 2015

%F 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

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

%t Table[Product[1 + Binomial[n,k], {k,0,n}], {n,0,15}] (* _Vaclav Kotesovec_, Oct 27 2017 *)

%o (PARI) { a(n) = prod(k=0,n, 1 + binomial(n,k))}

%o for(n=0,15,print1(a(n),", ")) \\ _Paul D. Hanna_, Oct 27 2017

%o (Magma)

%o A129824:= func< n | (&*[1 + Binomial(n,k): k in [0..n]]) >;

%o [A129824(n): n in [0..20]]; // _G. C. Greubel_, Apr 26 2024

%o (SageMath)

%o def A129824(n): return product(1 + binomial(n,k) for k in range(n+1))

%o [A129824(n) for n in range(21)] # _G. C. Greubel_, Apr 26 2024

%Y Cf. A001142, A055612.

%K easy,nonn

%O 0,1

%A _Henry Gould_, Jun 03 2007

%E Corrected and extended by _Vaclav Kotesovec_, Oct 27 2017