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A117148 Number of parts in all partitions of n in which no part occurs more than 3 times. 4

%I

%S 1,3,6,8,15,24,36,50,75,102,143,197,264,349,467,606,789,1016,1299,

%T 1656,2100,2634,3302,4117,5106,6306,7772,9523,11639,14185,17216,20839,

%U 25166,30280,36361,43551,52022,62004,73753,87510,103638,122507,144496,170133

%N Number of parts in all partitions of n in which no part occurs more than 3 times.

%C a(n) = sum(A117147(n,k), k>=1).

%H Vaclav Kotesovec, <a href="/A117148/b117148.txt">Table of n, a(n) for n = 1..15000</a> (terms 1..1000 from Alois P. Heinz)

%F G.f.: product(1+x^j+x^(2j)+x^(3j), j=1..infinity) * sum((x^j+2x^(2j)+3x^(3j)) / (1+x^j+x^(2j)+x^(3j)), j=1..infinity).

%F a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(1/4) * n^(1/4)). - _Vaclav Kotesovec_, May 27 2018

%e a(4) = 8 because the partitions of 4 in which no part occurs more than 3 times are [4], [3,1], [2,2] and [2,1,1] ([1,1,1,1] does not qualify) with a total of 1+2+2+3=8 parts.

%p g:=product(1+x^j+x^(2*j)+x^(3*j),j=1..55) *sum((x^j+2*x^(2*j)+3*x^(3*j))/ (1+x^j+x^(2*j)+x^(3*j)), j=1..55): gser:=series(g,x=0,53): seq(coeff(gser,x^n),n=1..50);

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],

%p add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 3))))

%p end:

%p a:= n-> b(n, n)[2]:

%p seq(a(n), n=1..50); # _Alois P. Heinz_, Jan 08 2013

%t b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, Sum[Function[{l}, {l[[1]], l[[2]] + l[[1]]*j}][b[n-i*j, i-1]], {j, 0, Min[n/i, 3]}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* _Jean-Fran├žois Alcover_, May 26 2015, after _Alois P. Heinz_ *)

%Y Cf. A001935, A117147.

%Y Column k=3 of A210485. - _Alois P. Heinz_, Jan 23 2013

%K nonn

%O 1,2

%A _Emeric Deutsch_, Mar 07 2006

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Last modified April 11 02:02 EDT 2021. Contains 342886 sequences. (Running on oeis4.)