A366175 The number of lit cells in weakly decreasing partitions of n when light shines from the north west and only the first column is lit. Here partitions are represented from left to right by columns of cells.

1, 2, 5, 7, 15, 22, 35, 54, 86, 115, 175, 248, 351, 480, 662, 890, 1229, 1622, 2154, 2820, 3718, 4814, 6269, 8048, 10303, 13086, 16648, 20998, 26540, 33196, 41509, 51607, 64086, 79162, 97769, 120213, 147587, 180401, 220173, 267697, 325211, 393778, 476237

Offset: 1

Author

Arnold Knopfmacher, Oct 03 2023

Keywords

• nonn

References

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

Crossrefs

Cf. A366069, A179862, A366157.

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; Rest[CoefficientList[Series[Sum[(r + 1)*q^(r + 1)*Sum[T[r, s], {s, 0, r}], {r, 0, nmax}], {q, 0, nmax}], q]] (* Vaclav Kotesovec, Oct 04 2023 *)

Formula

G.f.: Sum_{r>=0} (r+1)*q^(r+1)*T_q(r) where 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.

Extensions

a(14)-a(16) from Vaclav Kotesovec, Oct 04 2023

a(17)-a(43) from Alois P. Heinz, Oct 04 2023