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A114424
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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.
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1
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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
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OFFSET
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1,8
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REFERENCES
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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).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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 *) *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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