OFFSET
1,2
REFERENCES
A. Blecher, A. Knopfmacher, and M. E. Mays, Casting light on integer partitions, preprint.
FORMULA
G.f.: Sum_{k>=0} (P(q)-T_q(k))
where P(q) is the partition g.f. Product_{i>=1} 1/(1-q^i)
and T_q(k)=Sum_{s=0..k} t[k,s] with t[r,s]=q^s*Sum_{i=0..s} t[r-1,i]
and initial conditions t[1,1]=q; t[2,1]=q(1+q); t[r,0]=1; t[r,s]=0 for s>r.
a(n) <= n * A000041(n). - David A. Corneth, Oct 04 2023
MATHEMATICA
T[r_, s_] := If[s > r, 0, If[s == 0, 1, If[r == 1 && s == 1, q, If[r == 2 && s == 1, q*(1 + q), q^s*Sum[T[r - 1, i], {i, 0, s}]]]]]; nmax = 15; Do[Print[SeriesCoefficient[Sum[PartitionsP[n]*q^n - Sum[T[r, s], {s, 0, r}], {r, 0, n}], {q, 0, n}]], {n, 1, nmax}] (* Vaclav Kotesovec, Oct 04 2023 *)
PROG
(PARI) a(n) = {my(res = 0); forpart(p = n, res+=qlit(p)); res}
qlit(v) = {my(res = v[#v], h = v[#v]-1); forstep(i = #v-1, 1, -1, res+=max(0, v[i]-h); h = max(h, v[i])-1); res} \\ David A. Corneth, Oct 04 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Arnold Knopfmacher, Oct 02 2023
EXTENSIONS
a(13)-a(15) from Vaclav Kotesovec, Oct 04 2023
More terms from David A. Corneth, Oct 04 2023
STATUS
approved