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

1,2

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_{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.

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 *)

Crossrefs

Cf. A366069, A179862, A366157.

Sequence in context: A358878 A306956 A337655 * A111328 A244962 A022629

Adjacent sequences: A366172 A366173 A366174 * A366176 A366177 A366178

Keyword

nonn

Author

Arnold Knopfmacher, Oct 03 2023

Extensions

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

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

Status

approved