A240447 Number of partitions of 2n such that (sum of parts having multiplicity 1) = sum of all other parts.

1, 0, 1, 1, 3, 2, 8, 5, 18, 19, 39, 34, 105, 74, 183, 202, 381, 344, 818, 684, 1459, 1499, 2662, 2578, 5279, 4756, 8835, 9287, 15655, 15538, 28319, 27178, 46709, 49166, 78303, 80747, 135134, 134945, 216255, 231483, 353557, 369918, 581337, 600500, 915010, 987925

Offset

0,5

Comments

The number of partitions of 2n+1 such that (sum of parts having multiplicity 1) = sum of all other parts is 0; see the Mathematica program at A240448 for related sequences.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..650 (first 64 terms from Manfred Scheucher)

Manfred Scheucher, C Code

Example

a(6) counts these 8 partitions of 12:  633, 6222, 62211, 6111111, 5331, 52221, 4332, 42111111 .

Maple

f := proc(L, p)
    a := 0 ;
    for i in L do
        if i = p then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
sp1 := proc(L)
    a1 :=0 ;
    ao :=0 ;
    for i in L do
        if f(L, i) = 1 then
            a1 := a1+i;
        else
            ao := ao+i;
        end if;
    end do:
    if ( a1 = ao) then
        true;
    else
        false;
    end if;
end proc:
A240447 := proc(n)
    a := 0 ;
    for p in combinat[partition](2*n) do
        if sp1(p) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 06 2015
# second Maple program:
b:= proc(n, i, s) option remember; `if`(n=0, `if`(s=0, 1, 0),
      `if`(i<1 or s>0 and i*(i+1)/2<s or abs(s)>n, 0,
       add(b(n-i*j, i-1, s+`if`(j=1, -i, i*j)), j=0..n/i)))
    end:
a:= n-> b(2*n$2, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, May 31 2015

Mathematica

ColumnForm[t = Table[Select[IntegerPartitions[n], 2 Total[First[Transpose[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]]] &[Tally[#]] == n &], {n, 0, 30, 2}]] (* shows partitions *)
Map[Length, t] (* A240447 *)  (* Peter J. C. Moses, Apr 02 2014 *)
b[n_, i_, s_] := b[n, i, s] = If[n == 0, If[s == 0, 1, 0], If[i<1 || s>0 && i*(i+1)/2 < s || Abs[s] > n, 0, Sum[b[n-i*j, i-1, s + If[j == 1, -i, i*j]], {j, 0, n/i}]]]; a[n_] := b[2*n, 2*n, 0]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

Crossrefs

Cf. A240448, A240449, A240451, A240452.

Sequence in context: A336478 A322845 A307705 * A135992 A182638 A172084

Adjacent sequences: A240444 A240445 A240446 * A240448 A240449 A240450

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2014

Extensions

More terms from Manfred Scheucher, May 30 2015

Status

approved