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A182977 Total number of parts that are neither the smallest part nor the largest part in all partitions of n.

%I A182977 #37 Dec 13 2023 20:29:11
%S A182977 0,0,0,0,0,0,1,2,6,12,22,39,66,103,159,243,352,510,721,1011,1391,1903,
%T A182977 2557,3436,4549,5999,7824,10187,13132,16886,21544,27414,34657,43703,
%U A182977 54797,68558,85328,105963,131028,161664,198710
%N A182977 Total number of parts that are neither the smallest part nor the largest part in all partitions of n.
%H A182977 Alois P. Heinz, <a href="/A182977/b182977.txt">Table of n, a(n) for n = 0..1000</a>
%F A182977 a(n) = A006128(n) - A182978(n).
%F A182977 G.f.: g(x) = Sum_{i>=1} Sum_{j>=i+1} (Sum_{k=i+1..j-1} x^{i+j+k}/(1-x^k)/Product_{k=i..j}(1-x^k)). - _Emeric Deutsch_, Dec 25 2015
%F A182977 a(n) = Sum_{k>=0} k*A265249(n,k). - _Emeric Deutsch_, Dec 25 2015
%e A182977 For n = 6 the partitions of 6 are
%e A182977 6
%e A182977 5 + 1
%e A182977 4 + 2
%e A182977 4 + 1 + 1
%e A182977 3 + 3
%e A182977 3 + (2) + 1 .......... the "2" is the part that counts.
%e A182977 3 + 1 + 1 + 1
%e A182977 2 + 2 + 2
%e A182977 2 + 2 + 1 + 1
%e A182977 2 + 1 + 1 + 1 + 1
%e A182977 1 + 1 + 1 + 1 + 1 + 1
%e A182977 There is only one part which is neither the smallest part nor the largest part in all partitions of 6, so a(6) = 1.
%p A182977 g := add(add((add(x^(i+j+k)/(1-x^k), k = i+1 .. j-1))/(mul(1-x^k, k = i .. j)), j = i+1 .. 80), i = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # _Emeric Deutsch_, Dec 25 2015
%Y A182977 Cf. A006128, A046746, A092269, A116686, A182978, A182984, A265249.
%K A182977 nonn
%O A182977 0,8
%A A182977 _Omar E. Pol_, Jul 17 2011
%E A182977 a(12) corrected and more terms a(13)-a(40) from _David Scambler_, Jul 18 2011