A194452 Total number of repeated parts in all partitions of n.

0, 0, 2, 3, 8, 12, 24, 35, 60, 87, 136, 192, 287, 396, 567, 773, 1074, 1439, 1958, 2587, 3454, 4514, 5931, 7666, 9951, 12736, 16341, 20743, 26354, 33184, 41807, 52262, 65329, 81144, 100721, 124344, 153390, 188303, 230940, 282063, 344100, 418242, 507762

Offset: 0

Author

Omar E. Pol, Nov 19 2011

Keywords

• nonn

Examples

			For n = 6 we have:
--------------------------------------
.                        Number of
Partitions             repeated parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 2
4 + 2 ...................... 0
2 + 2 + 2 .................. 3
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 4
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 24
So a(6) = 24.
		

Links

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

Crossrefs

Cf. A006128, A024786, A047967, A135010, A138121, A194544, A264405.

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]*j)]
         od; h
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 20 2011
g := add(x^(2*j)*(2-x^j)/(1-x^j), j = 1 .. 80)/mul(1-x^j, j = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Feb 02 2016

Mathematica

myCount[p_List] := Module[{t}, If[p == {}, 0, t = Transpose[Tally[p]][[2]]; Sum[If[t[[i]] == 1, 0, t[[i]]], {i, Length[t]}]]]; Table[Total[Table[myCount[p], {p, IntegerPartitions[i]}]], {i, 0, 20}] (* T. D. Noe, Nov 19 2011 *)
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]]*j]}]; h] ]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 25 2015, after Alois P. Heinz *)
Table[Length[Flatten[Select[Flatten[Split[#]&/@IntegerPartitions[n],1],Length[#]>1&]]],{n,0,60}] (* Harvey P. Dale, Jun 12 2024 *)

Formula

a(n) = A006128(n) - A024786(n+1).

a(n) = Sum_{k=2..n} k*A264405(n,k). - Alois P. Heinz, Dec 07 2015

G.f.: g = Sum_{j>0} (x^{2*j}*(2 - x^j)/(1-x^j))/Product_{k>0}(1 - x^k) (obtained by logarithmic differentiation of the bivariate g.f. given in A264405). - Emeric Deutsch, Feb 02 2016