A048574 Self-convolution of 1 2 3 5 7 11 15 22 30 42 56 77 ... (A000041).
1, 4, 10, 22, 43, 80, 141, 240, 397, 640, 1011, 1568, 2395, 3604, 5360, 7876, 11460, 16510, 23588, 33418, 47006, 65640, 91085, 125596, 172215, 234820, 318579, 430060, 577920, 773130, 1030007, 1366644, 1806445, 2378892, 3121835, 4082796
Offset: 2
Examples
a(4) = 22 because (1,2,3,5)*(5,3,2,1) = 5 + 6 + 6 + 5 = 22
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 2..5000
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 804
Crossrefs
Programs
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Haskell
a048574 n = a048574_list !! (n-2) a048574_list = f (drop 2 a000041_list) [1] where f (p:ps) rs = (sum $ zipWith (*) rs $ tail a000041_list) : f ps (p : rs) -- Reinhard Zumkeller, Nov 09 2015
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Maple
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 # second Maple program: a:= n-> (p-> add(p(j)*p(n-j), j=1..n-1))(combinat[numbpart]): seq(a(n), n=2..40); # Alois P. Heinz, May 26 2018 -
Mathematica
a[n_] := First[ ListConvolve[ pp = Array[ PartitionsP, n], pp]]; Table[ a[n], {n, 1, 36}] (* Jean-François Alcover, Oct 21 2011 *) Table[ListConvolve[PartitionsP[Range[n]],PartitionsP[Range[n]]],{n,40}]// Flatten (* Harvey P. Dale, Oct 29 2020 *) -
PARI
a(n) = sum(k=1, n-1, numbpart(k)*numbpart(n-k)); \\ Michel Marcus, Dec 11 2016
Formula
From Franklin T. Adams-Watters, Feb 08 2006: (Start)
G.f.: ((Product_{k>0} 1/(1-x^k))-1)^2 = (exp(Sum_{k>0} (x^k/(1-x^k)/k))-1)^2. (End)
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)). - Vaclav Kotesovec, Mar 10 2018
Extensions
More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000
Comments