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

0,3

Links

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A303851 A218542 A379646 * A242516 A282281 A321175

Adjacent sequences: A194449 A194450 A194451 * A194453 A194454 A194455

Keyword

nonn

Author

Omar E. Pol, Nov 19 2011

Status

approved