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

0, 0, 2, 3, 10, 14, 33, 46, 87, 125, 208, 291, 461, 633, 942, 1292, 1851, 2491, 3484, 4629, 6321, 8326, 11143, 14513, 19168, 24720, 32185, 41193, 53030, 67297, 85830, 108116, 136651, 171040, 214462, 266731, 332197, 410730, 508201, 625082, 768920, 940938

Offset: 0

Author

Omar E. Pol, Nov 19 2011

Keywords

• nonn

Examples

			For n = 6 we have:
--------------------------------------
.                          Sum of
Partitions             repeated parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 6
4 + 2 ...................... 0
2 + 2 + 2 .................. 6
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 6
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
--------------------------------------
Total ..................... 33
So a(6) = 33.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A066186, A103628, A135010, A138121, A194452.

Programs

Maple

b:= proc(n, i) option remember; local h, j, t;
      if n<0 then [0, 0]
    elif n=0 then [1, 0]
    elif i<1 then [0, 0]
    else h:= [0, 0];
         for j from 0 to iquo(n, i) do
           t:= b(n-i*j, i-1);
           h:= [h[1]+t[1], h[2]+t[2]+`if`(j<2, 0, t[1]*i*j)]
         od; h
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Nov 20 2011

Mathematica

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 *)
Table[Total[Flatten[Select[Flatten[Split/@IntegerPartitions[n],1], Length[ #]> 1&]]],{n,0,50}] (* Harvey P. Dale, Jan 24 2019 *)

Formula

a(n) = A066186(n) - A103628(n), n >= 1.

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

Extensions

More terms from Alois P. Heinz, Nov 20 2011