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%I #24 Nov 18 2020 19:40:55
%S 1,1,3,8,31,141,819,5562,43773,389203,3858136,42152116,503098359,
%T 6511429138,90824834615,1358130449902,21672033893102,367570633594883,
%U 6602838664294634,125228962373218571,2500582942246200527,52437202425839368049,1152133477802718430790
%N Sum of NumberOfParts!/NumberOfDifferentParts! for all integer partitions of n.
%H Alois P. Heinz, <a href="/A108492/b108492.txt">Table of n, a(n) for n = 0..450</a>
%F 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.
%F a(n) ~ n! * (1 + 1/(2*n) + 1/n^2 + 13/(6*n^3) + 19/(3*n^4) + 22/n^5 + 2057/(24*n^6)). - _Vaclav Kotesovec_, Sep 06 2015
%e a(4) = 31 because n=4 has the following A000041(4) = 5 partitions:
%e i=1: (1111) with p(1,4)=4, d(1,4)=1; 4!/1! = 24;
%e i=2: (112) with p(2,4)=3, d(2,4)=2; 3!/2! = 3;
%e i=3: (13) with p(3,4)=2, d(3,4)=2; 2!/2! = 1;
%e i=4: (4) with p(4,4)=1, d(4,4)=1; 1!/1! = 1;
%e i=5: (22) with p(5,4)=2, d(5,4)=1; 2!/1! = 2;
%e Their contributions sum up to 24+3+1+1+2 = 31 = a(4).
%p A108492 := proc(n::integer) local i,prttnlst,prttn,ZahlTeile,liste,ZahlVerschiedenerTeile,A108492;
%p # 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
%p 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;
%p # Second Maple program:
%p b:= proc(n,i,l,p,d) option remember;
%p if n<0 then 0
%p elif n=0 then p!/d!
%p elif i=0 then 0
%p else b(n, i-1, l, p, d) +b(n-i, i, i, p+1, `if`(i=l, d, d+1))
%p fi
%p end:
%p a:= n-> b(n, n, 0, 0, 0):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 25 2011
%t b[n_, i_, l_, p_, d_] := b[n, i, l, p, d] = Which[n<0, 0, n==0, p!/d!, i==0, 0, True, b[n, i-1, l, p, d]+b[n-i, i, i, p+1, If[i==l, d, d+1]]];
%t a[n_] := b[n, n, 0, 0, 0];
%t a /@ Range[0, 30] (* _Jean-François Alcover_, Nov 18 2020, after _Alois P. Heinz_ *)
%Y Cf. A000041, A007318, A008277, A339006.
%K nonn
%O 0,3
%A _Thomas Wieder_, Jun 05 2005
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