cp's OEIS Frontend

A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.

%I A286335 #20 Sep 03 2019 15:23:46
%S A286335 1,1,0,1,1,0,1,2,1,0,1,3,3,2,0,1,4,6,6,2,0,1,5,10,13,9,3,0,1,6,15,24,
%T A286335 24,14,4,0,1,7,21,40,51,42,22,5,0,1,8,28,62,95,100,73,32,6,0,1,9,36,
%U A286335 91,162,206,190,120,46,8,0,1,10,45,128,259,384,425,344,192,66,10,0
%N A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.
%C A286335 A(n,k) is the number of partitions of n into distinct parts (or odd parts) with k types of each part.
%H A286335 Seiichi Manyama, <a href="/A286335/b286335.txt">Antidiagonals n = 0..139, flattened</a>
%H A286335 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A286335 G.f. of column k: Product_{j>=1} (1 + x^j)^k.
%F A286335 A(n,k) = Sum_{i=0..k} binomial(k,i) * A308680(n,k-i). - _Alois P. Heinz_, Aug 29 2019
%e A286335 A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).
%e A286335 Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).
%e A286335 Square array begins:
%e A286335   1,  1,  1,   1,   1,   1,  ...
%e A286335   0,  1,  2,   3,   4,   5,  ...
%e A286335   0,  1,  3,   6,  10,  15,  ...
%e A286335   0,  2,  6,  13,  24,  40,  ...
%e A286335   0,  2,  9,  24,  51,  95,  ...
%e A286335   0,  3, 14,  42, 100, 206,  ...
%p A286335 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A286335      (t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))
%p A286335     end:
%p A286335 A:= (n, k)-> b(n$2, k):
%p A286335 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Aug 29 2019
%t A286335 Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y A286335 Columns k=0-32 give: A000007, A000009, A022567-A022596.
%Y A286335 Rows n=0-2 give: A000012, A001477, A000217.
%Y A286335 Main diagonal gives A270913.
%Y A286335 Antidiagonal sums give A299106.
%Y A286335 Cf. A144064, A286352, A308680.
%K A286335 nonn,tabl
%O A286335 0,8
%A A286335 _Ilya Gutkovskiy_, May 07 2017