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A284593
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Square array read by antidiagonals: T(n,k) = the number of pairs of partitions of n and k respectively, such that each partition is composed of distinct parts and the pair of partitions have no part in common.
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6
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1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 2, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 2, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 6, 5, 6, 5, 6, 4, 5, 10, 12, 5, 5, 6, 5, 6, 6, 5, 6, 5, 5, 12, 15, 7, 6, 8, 7, 8, 8, 8, 7, 8, 6, 7, 15
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OFFSET
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0,7
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COMMENTS
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LINKS
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FORMULA
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O.g.f. Product_{j >= 1} (1 + x^j + y^j) = Sum_{n,k >= 0} T(n,k)*x^n*y^k (see Wilf, Example 7).
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EXAMPLE
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Square array begins
n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
0 | 1 1 1 2 2 3 4 5 6 8 10 12 15 18: A000009
1 | 1 0 1 1 1 2 2 3 3 5 5 7 8 10: A096765
2 | 1 1 0 1 2 2 2 3 4 5 6 7 9 11: A015744
3 | 2 1 1 2 2 3 4 6 6 8 9 12 15 18
4 | 2 1 2 2 2 3 5 5 7 9 10 14 15 19
5 | 3 2 2 3 3 6 6 8 9 12 16 19 22 28
6 | 4 2 2 4 5 6 8 9 11 16 18 22 27 33
7 | 5 3 3 6 5 8 9 14 16 20 23 29 34 41
...
T(3,7) = 6: the six pairs of partitions of 3 and 7 into distinct parts and with no parts in common are (3, 7), (3, 6 + 1), (3, 5 + 2), (3, 4 + 2 + 1), (2 + 1, 7) and (2 + 1, 4 + 3).
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MAPLE
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ser := taylor(taylor(mul(1 + x^j + y^j, j = 1..10), x, 11), y, 11):
convert(ser, polynom):
s := convert(%, polynom):
with(PolynomialTools):
for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+expand((x^i+1)*b(n-i, min(n-i, i-1)))))
end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
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MATHEMATICA
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nmax = 12; M = CoefficientList[#, y][[;; nmax+1]]& /@ (Product[1 + x^j + y^j, {j, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& // Expand);
T[n_, k_] := M[[n+1, k+1]];
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CROSSREFS
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Main diagonal gives 2*A108796 (for n>0).
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KEYWORD
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AUTHOR
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STATUS
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approved
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