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A240861
Number of partitions p of n into distinct parts not including the number of parts.
20
1, 0, 1, 1, 2, 2, 2, 4, 4, 5, 6, 9, 10, 12, 14, 18, 22, 26, 30, 36, 42, 51, 60, 70, 81, 94, 110, 128, 148, 172, 198, 226, 260, 298, 342, 390, 446, 508, 577, 654, 742, 840, 951, 1074, 1212, 1366, 1538, 1728, 1940, 2176, 2440, 2732, 3056, 3416, 3814, 4254
OFFSET
0,5
LINKS
FORMULA
a(n) = A000009(n) - A240855(n).
EXAMPLE
a(10) counts these 6 partitions: {10}, {9,1}, {7,3}, {7,2,1}, {6,4}, {5,4,1}.
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, [x^p, 0], (f-> [add(coeff(f[1], x, j)*x^j
, j=i+1..degree(f[1])), f[2]+coeff(f[1], x, i)])(
b(n-i, min(n-i, i-1), p+1))+b(n, i-1, p)))
end:
a:= n-> g(n)-b(n$2, 0)[2]:
seq(a(n), n=0..55); # Alois P. Heinz, Mar 14 2024
MATHEMATICA
z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; MemberQ[p, Length[p]]], {n, 0, z}] (* A240855 *)
Table[Count[f[n], p_ /; !MemberQ[p, Length[p]]], {n, 0, z}] (* A240861 *)
CROSSREFS
Sequence in context: A282562 A035682 A054543 * A029046 A035372 A035576
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 14 2014
EXTENSIONS
a(0) changed to 1 by Alois P. Heinz, Mar 14 2024
STATUS
approved