A240874 Number of partitions p of n into distinct parts such that max(p) < 2*min(p).

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 4, 5, 6, 6, 7, 7, 7, 8, 9, 10, 10, 11, 11, 12, 14, 14, 15, 17, 17, 18, 19, 20, 23, 24, 25, 26, 28, 29, 31, 34, 35, 37, 40, 42, 44, 46, 48, 51, 55, 58, 61, 64, 67, 70, 75, 77, 82, 87, 90, 96, 101, 105, 111

Offset

0,6

Links

John Tyler Rascoe, Table of n, a(n) for n = 0..200

Formula

G.f.: Sum_{i>0} Sum_{j>=i} q^((i/2)*(i+(2*j)-1)) * q_binomial(i-1,j-i). - John Tyler Rascoe, Mar 16 2024

Example

a(12) counts these 3 partitions:  {12}, {7,5}, {5,4,3}.

Mathematica

z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Min[p]], {n, 0, z}]  (* this sequence *)
Table[Count[f[n], p_ /; Max[p] == 2*Min[p]], {n, 0, z}] (* A241035 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Min[p]], {n, 0, z}] (* A241036 *)
Table[Count[f[n], p_ /; Max[p] > 2*Min[p]], {n, 0, z}]  (* A241037 *)

Prog

(PARI)
p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {p_q(N+M)/(p_q(M)*p_q(N)); }
A_q(N) = {my(q='q+O('q^N), g=sum(i=1, N, sum(j=i, N-(i*(i+1)/2), q^((i/2)*(i+(2*j)-1)) * GB_q(i-1, j-i))));
concat([0], Vec(g))}
A_q(71) \\ John Tyler Rascoe, Mar 16 2024

Crossrefs

Cf. A241035, A241036, A241037.

Sequence in context: A235645 A325357 A294107 * A029379 A058776 A233296

Adjacent sequences: A240871 A240872 A240873 * A240875 A240876 A240877

Keyword

nonn,easy

Author

Clark Kimberling, Apr 15 2014

Status

approved