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A284593 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.

%I A284593 #30 Apr 10 2025 06:47:57
%S A284593 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,
%T A284593 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,
%U A284593 5,5,6,5,6,6,5,6,5,5,12,15,7,6,8,7,8,8,8,7,8,6,7,15
%N A284593 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.
%C A284593 Compare with A284592.
%H A284593 Alois P. Heinz, <a href="/A284593/b284593.txt">Antidiagonals n = 0..200, flattened</a>
%H A284593 H. S. Wilf, <a href="https://www.math.upenn.edu/~wilf/PIMS/PIMSLectures.pdf">Lectures on Integer Partitions</a>
%F A284593 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).
%F A284593 Antidiagonal sums are A032302.
%e A284593 Square array begins
%e A284593   n\k| 0  1  2  3  4  5  6   7   8   9  10  11  12  13
%e A284593 - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e A284593   0  | 1  1  1  2  2  3  4   5   6   8  10  12  15  18: A000009
%e A284593   1  | 1  0  1  1  1  2  2   3   3   5   5   7   8  10: A096765
%e A284593   2  | 1  1  0  1  2  2  2   3   4   5   6   7   9  11: A015744
%e A284593   3  | 2  1  1  2  2  3  4   6   6   8   9  12  15  18
%e A284593   4  | 2  1  2  2  2  3  5   5   7   9  10  14  15  19
%e A284593   5  | 3  2  2  3  3  6  6   8   9  12  16  19  22  28
%e A284593   6  | 4  2  2  4  5  6  8   9  11  16  18  22  27  33
%e A284593   7  | 5  3  3  6  5  8  9  14  16  20  23  29  34  41
%e A284593   ...
%e A284593 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).
%p A284593 # A284593 as a square array
%p A284593 ser := taylor(taylor(mul(1 + x^j + y^j, j = 1..10), x, 11), y, 11):
%p A284593 convert(ser, polynom):
%p A284593 s := convert(%, polynom):
%p A284593 with(PolynomialTools):
%p A284593 for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
%p A284593 # second Maple program:
%p A284593 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A284593       b(n, i-1)+expand((x^i+1)*b(n-i, min(n-i, i-1)))))
%p A284593     end:
%p A284593 T:= (n, k)-> coeff(b(n+k$2), x, k):
%p A284593 seq(seq(T(n, d-n), n=0..d), d=0..14);  # _Alois P. Heinz_, Aug 24 2019
%t A284593 nmax = 12; M = CoefficientList[#, y][[;; nmax+1]]& /@ (Product[1 + x^j + y^j, {j, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& // Expand);
%t A284593 T[n_, k_] := M[[n+1, k+1]];
%t A284593 Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 07 2019 *)
%Y A284593 Rows n=0..2 give A000009, A096765, A015744.
%Y A284593 Main diagonal gives A365662.
%Y A284593 Antidiagonal sums give A032302.
%Y A284593 Cf. A284592, A322210.
%K A284593 nonn,tabl,easy
%O A284593 0,7
%A A284593 _Peter Bala_, Mar 30 2017