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A227926 Triangle read by rows: number of pyramid polycubes counted by height and volume.

%I A227926 #22 Jul 20 2020 13:09:21
%S A227926 1,2,1,2,4,1,3,8,4,1,2,16,10,4,1,4,22,22,10,4,1,2,36,40,24,10,4,1,4,
%T A227926 47,66,46,24,10,4,1,3,60,110,84,48,24,10,4,1,4,83,158,144,90,48,24,10,
%U A227926 4,1,2,100,233,232,162,92,48,24,10,4,1,6,116,327,357,266,168,92,48,24,10,4,1,2,148,444,544,435,284,170,92,48,24,10,4,1
%N A227926 Triangle read by rows: number of pyramid polycubes counted by height and volume.
%C A227926 A pyramid polycube is obtained by gluing together horizontal plateaux (parallelepipeds of height 1) in such a way that (0,0,0) belongs to the first plateau and each cell with coordinate (0,b,c) belonging to the first plateau is such that b, c >= 0. If the cell with coordinates (a,b,c) belongs to the (a+1)-st plateau (a>0), then the cell with coordinates (a-1, b, c) belongs to the a-th plateau.
%H A227926 David Goodger, <a href="http://puzzler.sourceforge.net/docs/polycubes.html">Polycubes: Puzzles & Solutions</a>
%F A227926 The number n_{i,j,h,v} of pyramids of volume v, height h and such that the highest plateau has volume i * j is given by the recurrence: n_{i,j,h,v} = Sum_{a=0..(i*j*h-v)/((h-1)*j)} Sum_{b=0..(j*(h*(i+a)-a)-v)/((i+a)*(k-1))} (a+1)*(b+1)*n_{i+a,j+a,h-1,v-i*j}.
%p A227926 calcPyr:=proc(i,j,k,l) option remember;
%p A227926 if (l<0) then 0
%p A227926 elif (i*j*k>l) then 0
%p A227926 elif k=1 then if (i*j=l) then 1 else 0;fi;
%p A227926 else s:=0; a:=0;b:=0;
%p A227926 while ((i+a)*j*(k-1)<=l-i*j) do
%p A227926 b:=0;
%p A227926 while ((i+a)*(j+b)*(k-1)<=l-i*j) do
%p A227926 s:=s+(a+1)*(b+1)*calcPyr(i+a,j+b,k-1,l-i*j);
%p A227926 b:=b+1;
%p A227926 od;
%p A227926 a:=a+1;
%p A227926 od;
%p A227926 s;
%p A227926 fi;
%p A227926 end;
%p A227926 countPyr:=proc(l)
%p A227926 s:=0;
%p A227926 for k to l do
%p A227926 i:=1:
%p A227926 while (i*k<=l) do
%p A227926 j:=1;
%p A227926 while (i*k*j<=l) do
%p A227926 s:=s+t^k*calcRecPyr(i,j,k,l);
%p A227926 j:=j+1;
%p A227926 od:
%p A227926 i:=i+1;
%p A227926 od;
%p A227926 od;
%p A227926 s;
%p A227926 end;
%p A227926 [1,seq(countPyr(ii),ii=1..200)];
%Y A227926 The numbers of pyramids counted by volume are given by A229914.
%K A227926 nonn,tabl
%O A227926 1,2
%A A227926 _Matthieu Deneufchâtel_, Oct 09 2013