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A118806 Triangle read by rows: T(n,k) is the number of partitions of n having k parts of multiplicity 3 (n,k>=0).

%I A118806 #5 Mar 30 2012 17:36:09
%S A118806 1,1,2,2,1,5,6,1,9,2,12,3,19,3,24,5,1,34,8,43,13,62,13,2,77,23,1,105,
%T A118806 28,2,132,40,4,177,49,5,220,71,6,287,89,8,1,356,123,11,462,147,18,570,
%U A118806 198,23,1,723,249,29,1,888,329,37,1,1121,400,50,4,1370,518,69,1,1705,642,85
%N A118806 Triangle read by rows: T(n,k) is the number of partitions of n having k parts of multiplicity 3 (n,k>=0).
%C A118806 T(n,0)=A118807(n). T(n,1)=A118808(n). Row sums yield the partition numbers (A000041). Sum(k*T(n,k), k>=0)=A117524(n) (n>=1).
%H A118806 Alois P. Heinz, <a href="/A118806/b118806.txt">Rows n = 0..703</a>
%F A118806 G.f.: product(1+x^j+x^(2j)+tx^(3j)+x^(4j)/(1-x^j), j=1..infinity).
%e A118806 T(12,2) = 2 because we have [3,3,3,1,1,1] and [3,2,2,2,1,1,1].
%e A118806 Triangle starts:
%e A118806 1;
%e A118806 1;
%e A118806 2;
%e A118806 2,  1;
%e A118806 5;
%e A118806 6,  1;
%e A118806 9,  2;
%e A118806 12, 3;
%p A118806 g:=product(1+x^j+x^(2*j)+t*x^(3*j)+x^(4*j)/(1-x^j),j=1..35): gser:=simplify(series(g,x=0,35)): P[0]:=1: for n from 1 to 30 do P[n]:=coeff(gser,x^n) od: for n from 0 to 30 do seq(coeff(P[n],t,j),j=0..degree(P[n])) od; # sequence given in triangular form
%Y A118806 Cf. A000041, A118807, A118808, A117524, A116595, A116644.
%K A118806 nonn,tabf
%O A118806 0,3
%A A118806 _Emeric Deutsch_, Apr 29 2006