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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.

%I A366157 #27 Oct 05 2023 02:54:52
%S A366157 1,4,8,17,27,49,74,118,174,263,371,540,747,1048,1429,1954,2610,3513,
%T A366157 4631,6123,7978,10398,13397,17277,22054,28131,35605,45004,56502,70879,
%U A366157 88370,110033,136325,168612,207637,255308,312689,382373,466004,566979,687685,832793,1005654
%N 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.
%D A366157 A. Blecher, A. Knopfmacher, and M. E. Mays, Casting light on integer partitions, preprint.
%F A366157 G.f.: Sum_{k>=0} (P(q)-T_q(k))
%F A366157 where P(q) is the partition g.f. Product_{i>=1} 1/(1-q^i)
%F A366157 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]
%F A366157 and initial conditions t[1,1]=q; t[2,1]=q(1+q); t[r,0]=1; t[r,s]=0 for s>r.
%F A366157 a(n) <= n * A000041(n). - _David A. Corneth_, Oct 04 2023
%t A366157 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 *)
%o A366157 (PARI) a(n) = {my(res = 0); forpart(p = n, res+=qlit(p)); res}
%o A366157 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
%Y A366157 Cf. A000041, A366069, A366175.
%K A366157 nonn
%O A366157 1,2
%A A366157 _Arnold Knopfmacher_, Oct 02 2023
%E A366157 a(13)-a(15) from _Vaclav Kotesovec_, Oct 04 2023
%E A366157 More terms from _David A. Corneth_, Oct 04 2023