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A129824 a(n) = Product_{k=0..n} (1 + binomial(n,k)). 2
2, 4, 12, 64, 700, 17424, 1053696, 160579584, 62856336636, 63812936890000, 168895157342195152, 1169048914836855865344, 21209591746609937928524800, 1010490883477487017627972550656, 126641164340871500483202065902080000, 41817338589698457759723104703370865147904 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

A product analog of the binomial expansion.

The sequence is a special case of a(n) = Product_{k=0..n} (1 + C(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

REFERENCES

H. W. Gould, A product analog of the binomial expansion, unpublished manuscript, Jun 03 2007.

LINKS

Table of n, a(n) for n=0..15.

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

for(n=0, 15, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 27 2017

CROSSREFS

Cf. A001142.

Sequence in context: A136512 A137160 A217716 * A266463 A013207 A172165

Adjacent sequences:  A129821 A129822 A129823 * A129825 A129826 A129827

KEYWORD

easy,nonn

AUTHOR

Henry Gould, Jun 03 2007

EXTENSIONS

Corrected and extended by Vaclav Kotesovec, Oct 27 2017

STATUS

approved

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Last modified March 4 04:39 EST 2021. Contains 341779 sequences. (Running on oeis4.)