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A114087 Triangle read by rows: T(n,k) is the number of partitions of n whose tails below their Durfee squares have size k (n>=1; 0<=k<=n-1).

%I A114087 #14 May 24 2021 14:55:38
%S A114087 1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,3,2,3,1,1,1,3,3,3,3,1,1,1,4,3,5,3,4,1,
%T A114087 1,1,5,4,5,5,4,4,1,1,1,6,5,7,5,7,4,5,1,1,1,7,6,9,7,7,7,5,5,1,1,1,9,7,
%U A114087 11,10,10,7,9,5,6,1,1,1,10,9,13,12,14,10,9,9,6,6,1,1,1,12,10,17,15,17,15
%N A114087 Triangle read by rows: T(n,k) is the number of partitions of n whose tails below their Durfee squares have size k (n>=1; 0<=k<=n-1).
%C A114087 Row sums yield A000041. Column 0 is A003114. Sum_{k=0..n-1} k*T(n,k) = A116365(n).
%D A114087 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
%D A114087 G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
%H A114087 Alois P. Heinz, <a href="/A114087/b114087.txt">Rows n = 1..141, flattened</a>
%F A114087 G.f.: Sum_(q^(k^2)/Product_((1-q^j)(1-(t*q)^j), j=1..k), k=1..infinity).
%e A114087 T(6,2) = 3 because we have [4,1,1], [2,2,2] and [2,2,1,1] (the bottom tails are [1,1], [2] and [1,1], respectively, each being a partition of 2).
%p A114087 g:=sum(z^(k^2)/product((1-z^j),j=1..k)/product((1-(t*z)^i),i=1..k),k=1..20): gserz:=simplify(series(g,z=0,30)): for n from 1 to 14 do P[n]:=coeff(gserz,z^n) od: for n from 1 to 14 do seq(coeff(t*P[n],t^j),j=1..n) od; # yields sequence in triangular form
%p A114087 # second Maple program:
%p A114087 b:= proc(n, i) option remember;
%p A114087       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A114087     end:
%p A114087 T:= (n, k)-> add(b(k, d)*b(n-d^2-k, d), d=0..floor(sqrt(n))):
%p A114087 seq(seq(T(n, k), k=0..n-1), n=1..20); # _Alois P. Heinz_, Apr 09 2012
%t A114087 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]] ]; T[n_, k_] := Sum[b[k, d]*b[n-d^2-k, d], {d, 0, Floor[Sqrt[n]]}]; Table[Table[ T[n, k], {k, 0, n-1}], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Feb 19 2015, after _Alois P. Heinz_ *)
%Y A114087 Cf. A000041, A003114, A115994, A115995, A116365, A114088, A114089.
%K A114087 nonn,tabl
%O A114087 1,7
%A A114087 _Emeric Deutsch_, Feb 12 2006