login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A194544 Total sum of repeated parts in all partitions of n. 6
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 * A075770 A135101 A108065

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 18 18:56 EDT 2019. Contains 328197 sequences. (Running on oeis4.)