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%I #18 Apr 05 2025 18:32:58
%S 1,1,2,3,6,12,20,48,120,240,540,1440,4320,11520,30240,64512,207360,
%T 725760,2419200,7257600,17418240,39191040,174182400,696729600,
%U 2786918400,9405849600,25082265600,65840947200,182891520000,1003290624000,4514807808000,21069103104000
%N Maximum possible product of differences of every pair in a set of nonnegative integers with sum n.
%C It seems that for n>=45, if m(m+1)/2<=n<(m+1)(m+2)/2, the set to provide the maximum product has m+1 elements, such as for n=46, the maximum product is reached by set {0,1,2,3,4,5,6,7,8,10}.
%H Chinese BBS, <a href="https://bbs.emath.ac.cn/thread-19972-1-1.html">Find the heap array with the largest product of the differences between the two pairs in 2025</a>
%e For n=7, the nonnegative integer set {0,1,2,4} has sum 7 and the product of number pairs is (1-0)*(2-0)*(4-0)*(2-1)*(4-1)*(4-2)=48 which is larger than any other sets with sum 7, so a(7)=48.
%Y Cf. A002620.
%K nonn
%O 0,3
%A _Zhao Hui Du_, Mar 21 2025