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A207033 Total number of parts >= 3 in all partitions of n.

%I A207033 #26 Mar 05 2021 21:43:50
%S A207033 0,0,1,2,4,8,13,22,35,54,80,121,172,247,347,484,661,906,1215,1632,
%T A207033 2162,2855,3730,4871,6290,8111,10381,13252,16802,21269,26750,33583,
%U A207033 41948,52277,64862,80326,99055,121922,149541,183052,223350,272038,330343,400450,484154
%N A207033 Total number of parts >= 3 in all partitions of n.
%H A207033 Alois P. Heinz, <a href="/A207033/b207033.txt">Table of n, a(n) for n = 1..1000</a>
%F A207033 G.f.: Sum_{k>=1} x^(3*k)/(1 - x^k) / Product_{j>=1} (1 - x^j). - _Ilya Gutkovskiy_, Mar 05 2021
%e A207033 a(4) = 2, because 2 parts have size >= 3 in all partitions of 4: [1,1,1,1], [1,1,2], [2,2], [1,3], [4].
%p A207033 b:= proc(n, i) option remember; local f, g;
%p A207033       if n=0 then [1, 0]
%p A207033     elif i<1 then [0, 0]
%p A207033     elif i>n then b(n, i-1)
%p A207033     else f:= b(n, i-1); g:= b(n-i, i);
%p A207033          [f[1]+g[1], f[2]+g[2] +`if`(i>2, g[1], 0)]
%p A207033       fi
%p A207033     end:
%p A207033 a:= n-> b(n, n)[2]:
%p A207033 seq(a(n), n=1..50);  # _Alois P. Heinz_, Feb 19 2012
%t A207033 b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i < 1, {0, 0}, i > n, b[n, i - 1], True, f = b[n, i - 1]; g = b[n - i, i]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i > 2, g[[1]], 0]}]];
%t A207033 a[n_] := b[n, n][[2]];
%t A207033 Array[a, 50] (* _Jean-François Alcover_, Nov 12 2020, after _Alois P. Heinz_ *)
%Y A207033 Column 3 of A181187.
%Y A207033 Cf. A006128, A096541, A206563.
%K A207033 nonn
%O A207033 1,4
%A A207033 _Omar E. Pol_, Feb 18 2012
%E A207033 More terms from _Alois P. Heinz_, Feb 18 2012