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%I #16 Nov 29 2017 03:34:25
%S 1,2,-6,-12,-240,-65280,-4294901760,-18446744069414584320,
%T -340282366920938463444927863358058659840,
%U -115792089237316195423570985008687907852929702298719625575994209400481361428480
%N a(1)=1, a(2)=2. a(n), for every positive integer n, is such that Product_{k=1..n} (Sum_{j=1..k} a(j)) = Sum_{k=1..n} Product_{j=1..k} a(j).
%C There are an infinite number of sequences {a(k)}, with different values for a(1) and a(2) (a(1) must be 0 or 1; a(2) can be anything), where Product_{k=1..n} (Sum_{j=1..k} a(j)) = Sum_{k=1..n} Product_{j=1..k} a(j), for all positive integers n. Setting a(1) to 1 and a(2) to 2 results in the sequence here.
%C All sequences (not necessarily integer sequences) with a(1) = 0 trivially have the property in the sequence name because each product is zero. For a general sequence in this family with a(1) = 1 and a(2) any integer, then a(3) = -a(2)^2 - a(2) and, for n >= 4, a(n) = -a(2)^(2^(n-3))*(a(2)^(2^(n-3))-1), so that all terms after a(2) are negatives of oblong (or promic) numbers (A002378). - _Rick L. Shepherd_, Aug 10 2014
%H Rick L. Shepherd, <a href="/A132076/b132076.txt">Table of n, a(n) for n = 1..13</a>
%F For n >= 4, a(n) = -2^(2^(n-3)) * (2^(2^(n-3)) - 1).
%F For n >= 4, a(n) = -A002378(A051179(n-3)). - _Rick L. Shepherd_, Aug 10 2014
%e For n = 4, we have a(1) * (a(1)+a(2)) * (a(1)+a(2)+a(3)) * (a(1)+a(2)+a(3)+a(4)) = a(1) + a(1)*a(2) + a(1)*a(2)*a(3) + a(1)*a(2)*a(3)*a(4) =
%e 1 * (1+2) * (1+2-6) * (1+2-6-12) = 1 + 1*2 + 1*2*(-6) + 1*2*(-6)*(-12) = 135.
%o (PARI)
%o a(n) = if(n<1, ,if(n<3, n, if(n==3, -6, -2^(2^(n-3))*(2^(2^(n-3))-1)))) \\ _Rick L. Shepherd_, Aug 10 2014
%Y Cf. A002378, A051179.
%K easy,sign
%O 1,2
%A _Leroy Quet_, Oct 30 2007
%E More terms from _Max Alekseyev_, Apr 29 2010
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