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A194544 Total sum of repeated parts in all partitions of n.

%I A194544 #31 Jan 24 2019 16:23:04
%S A194544 0,0,2,3,10,14,33,46,87,125,208,291,461,633,942,1292,1851,2491,3484,
%T A194544 4629,6321,8326,11143,14513,19168,24720,32185,41193,53030,67297,85830,
%U A194544 108116,136651,171040,214462,266731,332197,410730,508201,625082,768920,940938
%N A194544 Total sum of repeated parts in all partitions of n.
%H A194544 Alois P. Heinz, <a href="/A194544/b194544.txt">Table of n, a(n) for n = 0..1000</a>
%F A194544 a(n) = A066186(n) - A103628(n), n >= 1.
%F A194544 a(n) ~ exp(sqrt(2*n/3)*Pi) * (1/(4*sqrt(3))-3*sqrt(3)/(8*Pi^2)) * (1 - Pi*(135+2*Pi^2)/(24*(2*Pi^2-9)*sqrt(6*n))). - _Vaclav Kotesovec_, Nov 05 2016
%e A194544 For n = 6 we have:
%e A194544 --------------------------------------
%e A194544 .                          Sum of
%e A194544 Partitions             repeated parts
%e A194544 --------------------------------------
%e A194544 6 .......................... 0
%e A194544 3 + 3 ...................... 6
%e A194544 4 + 2 ...................... 0
%e A194544 2 + 2 + 2 .................. 6
%e A194544 5 + 1 ...................... 0
%e A194544 3 + 2 + 1 .................. 0
%e A194544 4 + 1 + 1 .................. 2
%e A194544 2 + 2 + 1 + 1 .............. 6
%e A194544 3 + 1 + 1 + 1 .............. 3
%e A194544 2 + 1 + 1 + 1 + 1 .......... 4
%e A194544 1 + 1 + 1 + 1 + 1 + 1 ...... 6
%e A194544 --------------------------------------
%e A194544 Total ..................... 33
%e A194544 So a(6) = 33.
%p A194544 b:= proc(n, i) option remember; local h, j, t;
%p A194544       if n<0 then [0, 0]
%p A194544     elif n=0 then [1, 0]
%p A194544     elif i<1 then [0, 0]
%p A194544     else h:= [0, 0];
%p A194544          for j from 0 to iquo(n, i) do
%p A194544            t:= b(n-i*j, i-1);
%p A194544            h:= [h[1]+t[1], h[2]+t[2]+`if`(j<2, 0, t[1]*i*j)]
%p A194544          od; h
%p A194544       fi
%p A194544     end:
%p A194544 a:= n-> b(n, n)[2]:
%p A194544 seq(a(n), n=0..50); # _Alois P. Heinz_, Nov 20 2011
%t A194544 b[n_, i_] := b[n, i] = Module[{h, j, t}, Which [n<0, {0, 0}, n==0, {1, 0}, i<1, {0, 0}, True, h = {0, 0}; For[j=0, j <= Quotient[n, i], j++, t = b[n - i*j, i-1]; h = {h[[1]] + t[[1]], h[[2]] + t[[2]] + If[j<2, 0, t[[1]]* i*j]}]; h]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jan 15 2016, after _Alois P. Heinz_ *)
%t A194544 Table[Total[Flatten[Select[Flatten[Split/@IntegerPartitions[n],1], Length[ #]> 1&]]],{n,0,50}] (* _Harvey P. Dale_, Jan 24 2019 *)
%Y A194544 Cf. A066186, A103628, A135010, A138121, A194452.
%K A194544 nonn
%O A194544 0,3
%A A194544 _Omar E. Pol_, Nov 19 2011
%E A194544 More terms from _Alois P. Heinz_, Nov 20 2011