A188431 The number of n-full sets, F(n).

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 1, 2, 1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 10, 13, 14, 17, 20, 25, 28, 34, 40, 46, 54, 62, 69, 80, 90, 102, 115, 131, 144, 167, 186, 213, 239, 273, 304, 349, 388, 441, 495, 563, 625, 710, 790, 890, 990, 1114, 1232, 1387, 1530, 1713, 1894, 2119, 2330, 2605, 2866, 3192, 3512, 3910, 4289, 4774, 5237, 5809, 6377, 7068, 7739

Offset: 0

Author

Madjid Mirzavaziri, Mar 31 2011

Keywords

• nonn

Comments

Let A be a set of positive integers. We say that A is n-full if (sum A)=[n] for a positive integer n, where (sum A) is the set of all positive integers which are a sum of distinct elements of A and [n]={1,2,...,n}. Then F(n) denotes the number of n-full sets.

Also the number of distinct and complete partitions of n, by definition, which are counted by A000009 and A126796. - George Beck, Nov 06 2017

An integer partition of n is complete (see also A325781) if every number from 0 to n is the sum of some submultiset of the parts. The Heinz numbers of these partitions are given by A325986. - Gus Wiseman, May 31 2019

Examples

			a(26) = 10, because there are 10 26-full sets: {1,2,4,5,6,8}, {1,2,3,5,7,8}, {1,2,3,5,6,9}, {1,2,3,4,7,9}, {1,2,3,4,6,10}, {1,2,3,4,5,11}, {1,2,4,8,11}, {1,2,4,7,12}, {1,2,4,6,13}, {1,2,3,7,13}.
G.f.: 1 = 1/(1+x) + 1*x/((1+x)*(1+x^2)) + 0*x^2/((1+x)*(1+x^2)*(1+x^3)) + 1*x^3/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) +...+ a(n)*x^n / Product_{k=1..n+1} (1+x^k) +...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..1000 from Reinhard Zumkeller)
• Mohammad Saleh Dinparvar, Python program
• L. Naranjani and M. Mirzavaziri, Full Subsets of N, Journal of Integer Sequences, 14 (2011), Article 11.5.3.
• Arash Sal Moslehian, JavaScript p5 program for three different algorithms

Crossrefs

Cf. A188429, A188430, A126796.

Cf. A002033, A103295, A108917, A276024, A325763, A325765, A325781, A325782, A325788, A325986, A325790, A325791.

Programs

Haskell

import Data.MemoCombinators (memo2, integral, Memo)
a188431 n = a188431_list !! (n-1)
a188431_list = map
   (\x -> sum [fMemo x i | i <- [a188429 x .. a188430 x]]) [1..] where
   fMemo = memo2 integral integral f
   f _ 1 = 1
   f m i = sum [fMemo (m - i) j |
                j <- [a188429 (m - i) .. min (a188430 (m - i)) (i - 1)]]
-- Reinhard Zumkeller, Aug 06 2015

Maple

sums:= proc(s) local i, m;
          m:= max(s[]);
         `if`(m<1, {}, {m, seq([i, i+m][], i=sums(s minus {m}))})
       end:
a:= proc(n) local b;
      b:= proc(i,s) local si;
            if i=1 then `if`(sums(s)={$1..n}, 1, 0)
          else si:= s union {i};
               b(i-1, s)+ `if`(max(sums(si)[])>n, 0, b(i-1, si))
            fi
          end; b(n, {1})
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 03 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2n or i>n-i+1, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 20 2017

Mathematica

Sums[s_] := Sums[s] = With[{m = Max[s]}, If[m < 1, {}, Union @ Flatten @ Join[{m}, Table[{i, i + m}, {i, Sums[s ~Complement~ {m}]}]]]];
a[n_] := Module[{b}, b[i_, s_] := b[i, s] = Module[{si}, If[i == 1, If[Sums[s] == Range[n], 1, 0], si = s ~Union~ {i}; b[i-1, s] + If[Max[ Sums[si]] > n, 0, b[i - 1, si]]]]; b[n, {1}]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 80}] (* Jean-François Alcover, Apr 12 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Union[Total/@Union[Subsets[#]]]==Range[0,n]&]],{n,30}] (* Gus Wiseman, May 31 2019 *)

PARI

/* As coefficients in g.f. */
{a(n)=local(A=[1]); for(i=1, n+1, A=concat(A,0); A[#A]=polcoeff(1 - sum(m=1,#A,A[m]*x^m/prod(k=1, m, 1+x^k +x*O(x^#A) )), #A) ); A[n+1]}
for(n=0, 50, print1(a(n),", ")) /* Paul D. Hanna, Mar 06 2012 */

Formula

F(n) = Sum_(i=L(n) .. U(n), F(n,i)), where F(n,i) = Sum_(j=L(n-i) .. min(U(n-i),i-1), F(n-i,j) ) and L(n), U(n) are defined in A188429 and A188430, respectively.

G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1+x^k), with a(0)=1. - Paul D. Hanna, Mar 08 2012

a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(3/4), where c = 0.03316508... - Vaclav Kotesovec, Oct 21 2019

Extensions

More terms from Alois P. Heinz, Apr 03 2011

a(0)=1 prepended by Alois P. Heinz, May 20 2017