A366157 The number of lit cells in weakly decreasing partitions of n when light shines from the north west. Here partitions are represented from left to right by columns of cells.

1, 4, 8, 17, 27, 49, 74, 118, 174, 263, 371, 540, 747, 1048, 1429, 1954, 2610, 3513, 4631, 6123, 7978, 10398, 13397, 17277, 22054, 28131, 35605, 45004, 56502, 70879, 88370, 110033, 136325, 168612, 207637, 255308, 312689, 382373, 466004, 566979, 687685, 832793, 1005654

Offset: 1

Author

Arnold Knopfmacher, Oct 02 2023

Keywords

• nonn

References

• A. Blecher, A. Knopfmacher, and M. E. Mays, Casting light on integer partitions, preprint.

Crossrefs

Cf. A000041, A366069, A366175.

Programs

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 *)

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

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

Extensions

a(13)-a(15) from Vaclav Kotesovec, Oct 04 2023

More terms from David A. Corneth, Oct 04 2023