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).

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, 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, 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

Offset: 1

Author

Emeric Deutsch, Feb 12 2006

Keywords

• nonn
• tabl

Comments

Row sums yield A000041. Column 0 is A003114. Sum_{k=0..n-1} k*T(n,k) = A116365(n).

Examples

			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).
		

References

• G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
• G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A000041, A003114, A115994, A115995, A116365, A114088, A114089.

Programs

Maple

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
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
T:= (n, k)-> add(b(k, d)*b(n-d^2-k, d), d=0..floor(sqrt(n))):
seq(seq(T(n, k), k=0..n-1), n=1..20); # Alois P. Heinz, Apr 09 2012

Mathematica

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 *)

Formula

G.f.: Sum_(q^(k^2)/Product_((1-q^j)(1-(t*q)^j), j=1..k), k=1..infinity).