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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 15 2014
STATUS
approved