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%I #16 Jul 12 2020 11:24:05
%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 + C(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%D H. W. Gould, A product analog of the binomial expansion, unpublished manuscript, Jun 03 2007.
%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
%Y Cf. A001142.
%K easy,nonn
%O 0,1
%A _Henry Gould_, Jun 03 2007
%E Corrected and extended by _Vaclav Kotesovec_, Oct 27 2017
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