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A240855
Number of partitions p of n into distinct parts including the number of parts.
8
0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 4, 3, 5, 6, 8, 9, 10, 12, 16, 18, 22, 25, 29, 34, 41, 48, 55, 64, 74, 84, 98, 114, 130, 150, 170, 195, 222, 252, 287, 328, 371, 420, 475, 536, 604, 682, 766, 862, 970, 1088, 1218, 1365, 1526, 1704, 1904, 2124, 2366, 2637, 2934
OFFSET
0,7
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 101 terms from John Tyler Rascoe)
FORMULA
a(n) = A000009(n) - A240861(n).
G.f.: Sum_{i>0} Sum_{j=1..i} q^((i*(i+1)/2) + j*(j-1)) * [j-1,i-j]_q, where [N,M]_q = Product_{j=1..N+M}(1-q^j) / (Product_{j=1..M}(1-q^j) * (Product_{j=1..N}(1-q^j))^2). - John Tyler Rascoe, Mar 13 2024
EXAMPLE
a(10) counts these 4 partitions: 82, 631, 532, 4321.
MAPLE
h:= (p, i)-> add(coeff(p, x, j)*x^j, j=i+1..degree(p)):
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, [0$2],
`if`(n=0, [x^p, 0], (g-> [h(g[1], i), g[2]])(b(n, i-1, p)+
(f-> f+[0, coeff(f[1], x, i)])(b(n-i, min(n-i, i-1), p+1)))))
end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..58); # 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}] (* this sequence *)
Table[Count[f[n], p_ /; !MemberQ[p, Length[p]]], {n, 0, z}] (* A240861 *)
PROG
(PARI)
p_q(k) = {prod(j=1, k, 1-q^j); }
mGB_q(N, M) = {p_q(N+M)/(p_q(M)*(p_q(N)^2)); }
A_q(N) = {my(q='q+O('q^N), g=sum(i=1, N, sum(j=1, i, q^((i*(i+1)/2)+(j*(j-1))) * mGB_q(j-1, i-j))));
concat([0], Vec(g)) }
A_q(50) \\ John Tyler Rascoe, Mar 13 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 14 2014
STATUS
approved