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A127954 Triangle, A097805 * A127648.

%I A127954 #19 Oct 15 2020 04:43:15
%S A127954 1,0,2,0,2,3,0,2,6,4,0,2,9,12,5,0,2,12,24,20,6,0,2,15,40,50,30,7,0,2,
%T A127954 18,60,100,90,42,8,0,2,21,84,175,210,147,56,9,0,2,24,112,280,420,392,
%U A127954 224,72,10
%N A127954 Triangle, A097805 * A127648.
%C A127954 Row sums = A045623 starting (1, 2, 5, 12, 28, 64, ...).
%C A127954 Dropping the first column gives A128710. - _Peter Bala_, Mar 05 2013
%C A127954 T(n,k) is the number of ways to place n unlabeled balls into 2 boxes, make compositions of the integer number of balls in each box so that the total number of parts in both compositions is k. - _Geoffrey Critzer_, Sep 21 2013
%F A127954 A097805 * A127648 as infinite lower triangular matrices.
%F A127954 E.g.f.: 1/(1 - y*(x/(1-x)))^2. - _Geoffrey Critzer_, Sep 21 2013
%F A127954 O.g.f.: (1+A001263(x,y))^2, - _Vladimir Kruchinin_, Oct 15 2020
%e A127954 First few rows of the triangle are:
%e A127954   1;
%e A127954   0, 2;
%e A127954   0, 2, 3;
%e A127954   0, 2, 6, 4;
%e A127954   0, 2, 9, 12, 5;
%e A127954   0, 2, 12, 24, 20, 6;
%e A127954   0, 2, 15, 40, 50, 30, 7;
%e A127954   ...
%e A127954 T(4,3)=12. Place 4 unlabeled balls into 2 labeled boxes then make compositions of the integer number of balls in each box so that there are a total of 3 parts.
%e A127954 /**** 3 ways since there are 3 compositions of 4 into 3 parts.
%e A127954 */***  2 ways 1;1+2  and 1;2+1
%e A127954 **/**  2 ways 2;1+1  and 1+1;2.
%e A127954 ***/*  2 ways as above.
%e A127954 ****/  3 ways as above.
%e A127954 3+2+2+2+3=12. - _Geoffrey Critzer_, Sep 21 2013
%t A127954 nn=10;a=x/(1-x);CoefficientList[Series[1/(1-y a)^2,{x,0,nn}],{x,y}]//Grid (* _Geoffrey Critzer_, Sep 21 2013 *)
%Y A127954 Cf. A001263, A097805, A127648, A128710.
%K A127954 nonn,tabl
%O A127954 1,3
%A A127954 _Gary W. Adamson_, Feb 09 2007