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A116599 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 2 (n>=0, 0<=k<=floor(n/2)).

%I A116599 #18 Oct 28 2017 21:36:53
%S A116599 1,1,1,1,2,1,3,1,1,4,2,1,6,3,1,1,8,4,2,1,11,6,3,1,1,15,8,4,2,1,20,11,
%T A116599 6,3,1,1,26,15,8,4,2,1,35,20,11,6,3,1,1,45,26,15,8,4,2,1,58,35,20,11,
%U A116599 6,3,1,1,75,45,26,15,8,4,2,1,96,58,35,20,11,6,3,1,1,121,75,45,26,15,8,4,2,1
%N A116599 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 2 (n>=0, 0<=k<=floor(n/2)).
%C A116599 Row n has 1 + floor(n/2) terms.
%C A116599 Row sums are the partition numbers (A000041).
%H A116599 G. C. Greubel, <a href="/A116599/b116599.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F A116599 T(n,0) = A027336(n), Sum_{k=0..floor(n/2)} k*T(n,k) = A024786(n).
%F A116599 Column k has g.f.: x^(2*k)/[(1-x)*Product_{j>=0} ((1-x^j))] (k=0,1,2,...).
%F A116599 G.f.: 1/[(1-x)*(1-t*x^2)*Product_{j>=3}( (1-x^j) )].
%F A116599 T(n,k) = p(n-2*k) - p(n-2*k-2) for k<=(n-2)/2;
%F A116599 T(n, floor(n/2))=1 (follows at once from the g.f.).
%e A116599 T(6,1)=3 because we have [4,2], [3,2,1] and [2,1,1,1,1].
%e A116599 Triangle starts:
%e A116599 1;
%e A116599 1;
%e A116599 1,1;
%e A116599 2,1;
%e A116599 3,1,1;
%e A116599 4,2,1;
%e A116599 6,3,1,1;
%e A116599 8,4,2,1;
%p A116599 with(combinat): T:=proc(n,k) if k=floor(n/2) then 1 elif k<=(n-2)/2 then numbpart(n-2*k)-numbpart(n-2*k-2) fi end: for n from 0 to 18 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A116599 nn = 20; p = Product[1/(1 - x^i), {i, 3, nn}]; f[list_] := Select[list, # > 0 &]; Map[f, CoefficientList[Series[p /(1 - x)/(1 - y x^2), {x, 0, nn}], {x, y}]] // Flatten  (* _Geoffrey Critzer_, Jan 22 2012 *)
%Y A116599 Cf. A000041, A027336, A024786.
%K A116599 nonn,tabf
%O A116599 0,5
%A A116599 _Emeric Deutsch_, Feb 18 2006
%E A116599 Keyword tabl changed to tabf by _Michel Marcus_, Apr 09 2013