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A290262 Irregular triangle read by rows: rows give the (negated) nonzero coefficients of t in each term of the inverse power product expansion of 1 - t * x/(1-x).

%I A290262 #18 Jun 14 2022 15:27:38
%S A290262 1,1,1,1,1,1,2,2,1,1,2,2,1,1,3,4,2,1,3,5,5,3,1,1,4,9,13,13,9,4,1,1,4,
%T A290262 9,13,13,9,4,1,1,5,14,25,30,24,12,3,1,5,15,30,42,42,30,15,5,1,1,6,21,
%U A290262 48,75,81,60,30,10,2
%N A290262 Irregular triangle read by rows: rows give the (negated) nonzero coefficients of t in each term of the inverse power product expansion of 1 - t * x/(1-x).
%C A290262 Row sums are A290261(n). A regular version is A290320.
%e A290262 Triangle begins:
%e A290262   1;
%e A290262   1,  1;
%e A290262   1,  1,
%e A290262   1,  2,  2,  1;
%e A290262   1,  2,  2,  1;
%e A290262   1,  3,  4,  2;
%e A290262   1,  3,  5,  5,  3,  1;
%e A290262   1,  4,  9, 13, 13,  9,  4,  1;
%e A290262   1,  4,  9, 13, 13,  9,  4,  1;
%e A290262   1,  5, 14, 25, 30, 24, 12,  3;
%e A290262   1,  5, 15, 30, 42, 42, 30, 15,  5,  1;
%e A290262   1,  6, 21, 48, 75, 81, 60, 30, 10,  2;
%t A290262 eptrees[n_]:=Prepend[Join@@Table[Tuples[eptrees/@y],{y,Rest[IntegerPartitions[n]]}],n];
%t A290262 eptrans[a_][n_]:=Sum[(-1)^Count[tree,_List,{0,Infinity}]*Product[a[i],{i,Flatten[{tree}]}],{tree,eptrees[n]}];
%t A290262 Table[DeleteCases[CoefficientList[-eptrans[-t&][n],t],0],{n,12}]
%Y A290262 Cf. A220418, A273866, A289501, A290261, A290320.
%K A290262 tabf,nonn
%O A290262 1,7
%A A290262 _Gus Wiseman_, Jul 24 2017