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

1,2

References

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

Links

Table of n, a(n) for n=1..43.

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

Cf. A000041, A366069, A366175.

Sequence in context: A298950 A213494 A092321 * A366069 A026353 A067773

Adjacent sequences: A366154 A366155 A366156 * A366158 A366159 A366160

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