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%I #6 Dec 31 2015 08:17:13
%S 1,2,3,5,7,10,1,13,2,17,4,1,20,8,2,26,11,4,1,29,17,8,2,35,24,13,4,1,
%T 39,33,19,8,2,48,39,30,13,4,1,48,56,41,21,8,2,60,64,57,32,13,4,1,61,
%U 83,75,47,21,8,2,74,94,100,65,34,13,4,1
%N Triangle read by rows: T(n,k) is the number of partitions of n having k parts strictly between the smallest and the largest part (n>=1, k>=0).
%C Number of entries in row n is floor((n-4)/2) (n>=4).
%C Sum of entries of row n = A000041(n) = number of partitions of n.
%C T(n,0) = A265250(n).
%C Sum(k*T(n,k), k>=0) = A182977(n).
%F G.f.: G(t,x) = Sum_{i>=1} x^i/(1-x^i) + Sum_{i>=1} Sum_{j>=i+1} x^(i+j)/(1-x^i)/(1-x^j)/Product_{k=i+1..j-1} (1-tx^k).
%e T(8,2) = 1 because among the 22 partitions of 8 only [3,2,2,1] has 2 parts strictly between the smallest and the largest part.
%e Triangle starts:
%e 1;
%e 2;
%e 3;
%e 5;
%e 7;
%e 10, 1;
%e 13, 2;
%p g := add(x^i/(1-x^i), i=1..80)+add(add(x^(i+j)/((1-x^i)*(1-x^j)*mul(1-t*x^k, k=i+1..j-1)),j=i+1..80),i=1..80): gser := simplify(series(g,x=0,23)): for n to 22 do P[n]:= sort(coeff(gser,x,n)) end do: for n to 22 do seq(coeff(P[n],t,k), k=0..degree(P[n])) end do; # yields sequence in triangular form
%Y CF. A000041, A182977, A265250.
%K nonn,tabf
%O 1,2
%A _Emeric Deutsch_, Dec 25 2015
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