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A129824
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a(n) = Product_{k=0..n} (1 + binomial(n,k)).
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3
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2, 4, 12, 64, 700, 17424, 1053696, 160579584, 62856336636, 63812936890000, 168895157342195152, 1169048914836855865344, 21209591746609937928524800, 1010490883477487017627972550656, 126641164340871500483202065902080000, 41817338589698457759723104703370865147904
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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H. W. Gould, A product analog of the binomial expansion, unpublished manuscript, Jun 03 2007.
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LINKS
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FORMULA
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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
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EXAMPLE
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a(4) = (1+1)(1+4)(1+6)(1+4)(1+1) = 2*5*7*5*2 = 700.
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MATHEMATICA
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Table[Product[1 + Binomial[n, k], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 27 2017 *)
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PROG
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(PARI) { a(n) = prod(k=0, n, 1 + binomial(n, k))}
(Magma)
A129824:= func< n | (&*[1 + Binomial(n, k): k in [0..n]]) >;
(SageMath)
def A129824(n): return product(1 + binomial(n, k) for k in range(n+1))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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