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

%I A117148 #23 May 27 2018 03:37:39
%S A117148 1,3,6,8,15,24,36,50,75,102,143,197,264,349,467,606,789,1016,1299,
%T A117148 1656,2100,2634,3302,4117,5106,6306,7772,9523,11639,14185,17216,20839,
%U A117148 25166,30280,36361,43551,52022,62004,73753,87510,103638,122507,144496,170133
%N A117148 Number of parts in all partitions of n in which no part occurs more than 3 times.
%C A117148 a(n) = sum(A117147(n,k), k>=1).
%H A117148 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 A117148 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 A117148 a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(1/4) * n^(1/4)). - _Vaclav Kotesovec_, May 27 2018
%e A117148 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 A117148 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 A117148 # second Maple program:
%p A117148 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A117148       add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 3))))
%p A117148     end:
%p A117148 a:= n-> b(n, n)[2]:
%p A117148 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jan 08 2013
%t A117148 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-François Alcover_, May 26 2015, after _Alois P. Heinz_ *)
%Y A117148 Cf. A001935, A117147.
%Y A117148 Column k=3 of A210485. - _Alois P. Heinz_, Jan 23 2013
%K A117148 nonn
%O A117148 1,2
%A A117148 _Emeric Deutsch_, Mar 07 2006