|
| |
|
|
A108492
|
|
Sum of NumberOfParts!/NumberOfDifferentParts! for all integer partitions of n.
|
|
1
| |
|
|
1, 1, 3, 8, 31, 141, 819, 5562, 43773, 389203, 3858136, 42152116, 503098359, 6511429138, 90824834615, 1358130449902, 21672033893102, 367570633594883, 6602838664294634, 125228962373218571, 2500582942246200527
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..170
|
|
|
FORMULA
| a(n) = Sum_{i=1..A000041(n)} p(i,n)!/d(i,n)! with p(i,n) = number of parts of the i-th partition of n and d(i,n) = number of different parts of the i-th partition of n.
|
|
|
EXAMPLE
| a(4) = 31 because n=4 has the following A000041(4) = 5 partitions:
i=1: (1111) with p(1,4)=4, d(1,4)=1; 4!/1! = 24;
i=2: (112) with p(2,4)=3, d(2,4)=2; 3!/2! = 3;
i=3: (13) with p(3,4)=2, d(3,4)=2; 2!/2! = 1;
i=4: (4) with p(4,4)=1, d(4,4)=1; 1!/1! = 1;
i=5: (22) with p(5,4)=2, d(5,4)=1; 2!/1! = 2;
Their contributions sum up to 24+3+1+1+2 = 31 = a(4).
|
|
|
MAPLE
| A108492 := proc(n::integer) local i, prttnlst, prttn, ZahlTeile, liste, ZahlVerschiedenerTeile, A108492;
# Procedure A108492 calculates the sequence A108492 for the integer partitions of n. prttn = an integer partition of n. See also http://www.thomas-wieder.privat.t-online.de/default.html
prttnlst:=partition(n); A108492 := 0; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; ZahlTeile := nops(prttn); liste := convert(prttn, multiset); ZahlVerschiedenerTeile := nops(liste); A108492 := A108492 + (ZahlTeile!/ZahlVerschiedenerTeile!); od; print(n, A108492); end proc;
##
b:= proc(n, i, l, p, d) option remember;
if n<0 then 0
elif n=0 then p!/d!
elif i=0 then 0
else b(n, i-1, l, p, d) +b(n-i, i, i, p+1, `if` (i=l, d, d+1))
fi
end:
a:= n-> b(n, n, 0, 0, 0):
seq (a(n), n=0..30); # Alois P. Heinz, Apr 25 2011
|
|
|
CROSSREFS
| Cf. A000041, A007318, A008277.
Sequence in context: A148900 A148901 A148902 * A003470 A176304 A180385
Adjacent sequences: A108489 A108490 A108491 * A108493 A108494 A108495
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 05 2005
|
| |
|
|
