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A114087
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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).
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4
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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
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OFFSET
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1,7
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COMMENTS
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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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G.f.: Sum_(q^(k^2)/Product_((1-q^j)(1-(t*q)^j), j=1..k), k=1..infinity).
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EXAMPLE
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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).
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MAPLE
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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))):
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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]]] ]; 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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