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A254430 Number of "feasible" partitions with n parts.

%I A254430 #36 Nov 05 2018 03:07:22
%S A254430 1,3,16,183,4804,299558,45834625,17696744699,17644374475261,
%T A254430 46279884666882734,324101360547203133793
%N A254430 Number of "feasible" partitions with n parts.
%C A254430 This sequence answers the question: "How many sellers can each be provided with a distinct set of n-part 'feasible' weights described in A254296?" It counts all the n-part "feasible" partitions of all the natural numbers from (3^(n-1)+1)/2 to (3^n-1)/2. Here n resembles m in A254296.
%H A254430 Md Towhidul Islam & Md Shahidul Islam, <a href="http://arxiv.org/abs/1502.07730">Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance</a>, arXiv:1502.07730 [math.CO], 2015.
%F A254430 a(n) = Sum_{p=(3^(n-1)+1)/2..(3^n-1)/2} A254296(p).
%e A254430 For n=2, we count 2nd through 4th values of A254296. So a(2)=1+1+1=3.
%e A254430 For n=3, we count 5th through 13th values from A254296. So a(3)= 2+2+3+2+2+2+1+1+1 = 16.
%e A254430 For n=4, a(4)= Sum of 14th through 40th terms of A254296, that is, 183.
%t A254430 okQ[v_] := Module[{s = 0}, For[i = 1, i <= Length[v], i++, If[v[[i]] > 2s + 1, Return[False], s += v[[i]]]]; Return[True]];
%t A254430 a254296[n_] := With[{k = Ceiling[Log[3, 2n]]}, Select[Reverse /@ IntegerPartitions[n, {k}], okQ] // Length];
%t A254430 a[n_] := Sum[a254296[p], {p, (3^(n-1) + 1)/2, (3^n - 1)/2}];
%t A254430 Array[a, 5] (* _Jean-François Alcover_, Nov 04 2018, after _Charles R Greathouse IV_ in A254296 *)
%Y A254430 Cf. A254296, A254431, A254432, A254433, A254435, A254436, A254437, A254438, A254439, A254440, A254442.
%K A254430 nonn
%O A254430 1,2
%A A254430 _Md. Towhidul Islam_, Jan 30 2015
%E A254430 a(9)-a(11) from _Md. Towhidul Islam_, Apr 18 2015