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A116687 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the parts that are smaller than the largest part is equal to k (n>=1, k>=0).

%I A116687 #9 Jan 23 2025 06:50:02
%S A116687 1,2,2,1,3,1,1,2,2,2,1,4,1,3,2,1,2,3,2,4,3,1,4,1,5,3,5,3,1,3,3,2,6,5,
%T A116687 6,4,1,4,2,5,3,9,6,8,4,1,2,3,4,7,5,11,9,9,5,1,6,1,5,5,10,7,15,11,11,5,
%U A116687 1,2,5,2,7,8,13,11,18,15,13,6,1,4,1,9,3,11,10,19,14,24,18,15,6,1,4,3,2,12,5
%N A116687 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the parts that are smaller than the largest part is equal to k (n>=1, k>=0).
%C A116687 Row 1 has one term; row n (n>=2) has n-1 terms.
%C A116687 Row sums yield the partition numbers (A000041).
%C A116687 T(n,0) = A000005(n) (number of divisors of n).
%C A116687 T(n,1) = A032741(n-1) (number of proper divisors of n-1).
%C A116687 Sum_{k=0..n-2} k*T(n,k) = A116688(n).
%F A116687 G.f.: Sum_{i>=1} x^i/((1-x^i)*Product_{j=1..i-1} (1-t^j*x^j)).
%e A116687 T(6,2) = 3 because we have [4,2], [4,1,1] and [2,2,1,1].
%e A116687 Triangle starts:
%e A116687   1;
%e A116687   2;
%e A116687   2,1;
%e A116687   3,1,1;
%e A116687   2,2,2,1;
%e A116687   4,1,3,2,1;
%e A116687   ...
%p A116687 g:=sum(x^i/(1-x^i)/product(1-(t*x)^j,j=1..i-1),i=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 15 do P[n]:=coeff(gser,x^n) od: 1; for n from 2 to 15 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form
%Y A116687 Cf. A000041, A000005, A032741, A116688.
%K A116687 nonn,tabf
%O A116687 1,2
%A A116687 _Emeric Deutsch_, Feb 23 2006