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%I #27 Feb 07 2021 11:57:22
%S 1,4,10,22,43,80,141,240,397,640,1011,1568,2395,3604,5360,7876,11460,
%T 16510,23588,33418,47006,65640,91085,125596,172215,234820,318579,
%U 430060,577920,773130,1030007,1366644,1806445,2378892,3121835,4082796
%N Self-convolution of 1 2 3 5 7 11 15 22 30 42 56 77 ... (A000041).
%C Number of proper partitions of n into parts of two kinds (i.e. both kinds must be present). - _Franklin T. Adams-Watters_, Feb 08 2006
%H Reinhard Zumkeller, <a href="/A048574/b048574.txt">Table of n, a(n) for n = 2..5000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=804">Encyclopedia of Combinatorial Structures 804</a>
%F From _Franklin T. Adams-Watters_, Feb 08 2006: (Start)
%F a(0) = 0, a(n) = A000712(n)-2*A000041(n) for n>0.
%F a(n) = Sum_{k=1..n-1} A000041(k)*A000041(n-k).
%F G.f.: ((Product_{k>0} 1/(1-x^k))-1)^2 = (exp(Sum_{k>0} (x^k/(1-x^k)/k))-1)^2. (End)
%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)). - _Vaclav Kotesovec_, Mar 10 2018
%e a(4) = 22 because (1,2,3,5)*(5,3,2,1) = 5 + 6 + 6 + 5 = 22
%p spec := [S,{C=Sequence(Z,1 <= card),B=Set(C,1 <= card),S=Prod(B,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20); # _Franklin T. Adams-Watters_, Feb 08 2006
%p # second Maple program:
%p a:= n-> (p-> add(p(j)*p(n-j), j=1..n-1))(combinat[numbpart]):
%p seq(a(n), n=2..40); # _Alois P. Heinz_, May 26 2018
%t a[n_] := First[ ListConvolve[ pp = Array[ PartitionsP, n], pp]]; Table[ a[n], {n, 1, 36}] (* _Jean-François Alcover_, Oct 21 2011 *)
%t Table[ListConvolve[PartitionsP[Range[n]],PartitionsP[Range[n]]],{n,40}]// Flatten (* _Harvey P. Dale_, Oct 29 2020 *)
%o (Haskell)
%o a048574 n = a048574_list !! (n-2)
%o a048574_list = f (drop 2 a000041_list) [1] where
%o f (p:ps) rs = (sum $ zipWith (*) rs $ tail a000041_list) : f ps (p : rs)
%o -- _Reinhard Zumkeller_, Nov 09 2015
%o (PARI) a(n) = sum(k=1, n-1, numbpart(k)*numbpart(n-k)); \\ _Michel Marcus_, Dec 11 2016
%Y Cf. A000041, A000712, A023626.
%Y Essentially the same as A052837.
%Y Cf. A122768.
%Y Column k=2 of A060642.
%K easy,nice,nonn
%O 2,2
%A _Alford Arnold_
%E More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000
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