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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A304825 Sum of binomial(Y(2,p), 2) over the partitions p of n, where Y(2,p) is the number of part sizes with multiplicity 2 or greater in p. 0
1, 1, 3, 4, 9, 12, 22, 30, 50, 68, 105, 142, 210, 281, 400, 531, 736, 967, 1311, 1707, 2274, 2935, 3851, 4930, 6389, 8116, 10402, 13121, 16658, 20872, 26275, 32719, 40880, 50613, 62807, 77343, 95389, 116874, 143331, 174789, 213251, 258903, 314367, 380079, 459462 (list; graph; refs; listen; history; text; internal format)
OFFSET

6,3

LINKS

Table of n, a(n) for n=6..50.

FORMULA

a(n) = (A301313(n) - A024788(n))/4.

G.f.: q^6 /((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j).

EXAMPLE

For a(8), we sum over the partitions of eight. For each partition p, we take binomial(Y(2,p),2): that is, the number of parts with multiplicity at least two choose 2.

8................B(0,2) = 0

7,1..............B(0,2) = 0

6,2..............B(0,2) = 0

6,1,1............B(1,2) = 0

5,3..............B(0,2) = 0

5,2,1............B(0,2) = 0

5,1,1,1..........B(1,2) = 0

4,4..............B(1,2) = 0

4,3,1............B(0,2) = 0

4,2,2............B(1,2) = 0

4,2,1,1..........B(1,2) = 0

4,1,1,1,1........B(1,2) = 0

3,3,2............B(1,2) = 0

3,3,1,1..........B(2,2) = 1

3,2,2,1..........B(1,2) = 0

3,2,1,1,1........B(1,2) = 0

3,1,1,1,1,1......B(1,2) = 0

2,2,2,2..........B(1,2) = 0

2,2,2,1,1........B(2,2) = 1

2,2,1,1,1,1......B(2,2) = 1

2,1,1,1,1,1,1....B(1,2) = 0

1,1,1,1,1,1,1,1..B(1,2) = 0

---------------------------

Total.....................3

MAPLE

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,

      binomial(`if`(n>1, 1, 0)+p, 2), add(

      b(n-i*j, i-1, `if`(j>1, 1, 0)+p), j=0..n/i))

    end:

a:= n-> b(n$2, 0):

seq(a(n), n=6..60);  # Alois P. Heinz, May 19 2018

MATHEMATICA

Array[Total[Binomial[Count[Split@#, _?(Length@# >= 2 &)], 2] & /@IntegerPartitions[#]] &, 50]

CROSSREFS

Cf. A024786, A302347.

Sequence in context: A025613 A097063 A293569 * A026476 A002513 A034418

Adjacent sequences:  A304822 A304823 A304824 * A304826 A304827 A304828

KEYWORD

nonn

AUTHOR

Emily Anible, May 19 2018

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 August 22 12:10 EDT 2019. Contains 326177 sequences. (Running on oeis4.)