A048574 - OEIS

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

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OFFSET

2,2

COMMENTS

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 2..5000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 804

FORMULA

From Franklin T. Adams-Watters, Feb 08 2006: (Start)

a(0) = 0, a(n) = A000712(n)-2*A000041(n) for n>0.

a(n) = Sum_{k=1..n-1} A000041(k)*A000041(n-k).

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

EXAMPLE

a(4) = 22 because (1,2,3,5)*(5,3,2,1) = 5 + 6 + 6 + 5 = 22

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 *)

PROG

(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