A114424 Number of partitions of n such that the size of the tail below the Durfee square is equal to the size of the tail to the right of the Durfee square.

1, 0, 1, 1, 1, 1, 1, 4, 2, 4, 2, 9, 5, 9, 10, 17, 17, 17, 26, 29, 50, 34, 65, 61, 102, 72, 146, 130, 201, 170, 266, 289, 387, 388, 491, 611, 677, 811, 899, 1260, 1225, 1630, 1619, 2355, 2270, 3086, 2970, 4361, 4147, 5524, 5555, 7625, 7609, 9681, 10202, 13085

Offset

1,8

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, Table of n, a(n) for n = 1..1000

Formula

a(n) = coefficient of (t^0)(z^n) in G(t,1/t,z), where G(t,s,z)=sum(z^(k^2)/product((1-(tz)^j)(1-(sz)^j),j=1..k),k=1..infinity) is the trivariate g.f. according to partition size (z), size of the tail below the Durfee square (t) and size of the tail to the right of the Durfee square (s).

Example

a(9) = 2 because we have [5,1,1,1,1] with both tails of size 4 and [3,3,3] with both tails of size 0.

Maple

g:=sum(z^(k^2)/product((1-(t*z)^j)*(1-(z/t)^j), j=1..k), k=1..10): gser:=simplify(series(g, z=0, 65)): 1, seq(coeff(numer(coeff(gser, z^n)), t^(n-1)), n=2..60);
# 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:
a:= proc(n) local r; add(`if`(irem(n-d^2, 2, 'r')=1, 0,
                          b(r, d)^2), d=1..floor(sqrt(n)))
    end:
seq(a(n), n=1..70);  # 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]]]]; a[n_] := Sum[If[{q, r} = QuotientRemainder[n-d^2, 2]; r==1, 0, b[q, d]^2], {d, 1, Floor[Sqrt[n]]}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

Crossrefs

Sequence in context: A278527 A010474 A064887 * A246819 A056158 A330093

Adjacent sequences: A114421 A114422 A114423 * A114425 A114426 A114427

Keyword

nonn

Author

Emeric Deutsch, Feb 12 2006

Status

approved