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A240447
Number of partitions of 2n such that (sum of parts having multiplicity 1) = sum of all other parts.
5
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 05 2014
EXTENSIONS
More terms from Manfred Scheucher, May 30 2015
STATUS
approved