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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.
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%I #17 Oct 04 2023 12:37:10

%S 1,2,5,7,15,22,35,54,86,115,175,248,351,480,662,890,1229,1622,2154,

%T 2820,3718,4814,6269,8048,10303,13086,16648,20998,26540,33196,41509,

%U 51607,64086,79162,97769,120213,147587,180401,220173,267697,325211,393778,476237

%N 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.

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

%F 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.

%t 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 *)

%Y Cf. A366069, A179862, A366157.

%K nonn

%O 1,2

%A _Arnold Knopfmacher_, Oct 03 2023

%E a(14)-a(16) from _Vaclav Kotesovec_, Oct 04 2023

%E a(17)-a(43) from _Alois P. Heinz_, Oct 04 2023