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

0,3

Links

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A171126 A299205 A268878 * A348475 A075770 A135101

Adjacent sequences: A194541 A194542 A194543 * A194545 A194546 A194547

Keyword

nonn

Author

Omar E. Pol, Nov 19 2011

Extensions

More terms from Alois P. Heinz, Nov 20 2011

Status

approved