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A286653
Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^(k*j))/(1 - x^j).
9
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 3, 4, 3, 0, 1, 1, 2, 3, 4, 5, 4, 0, 1, 1, 2, 3, 5, 6, 7, 5, 0, 1, 1, 2, 3, 5, 6, 9, 9, 6, 0, 1, 1, 2, 3, 5, 7, 10, 12, 13, 8, 0, 1, 1, 2, 3, 5, 7, 10, 13, 16, 16, 10, 0, 1, 1, 2, 3, 5, 7, 11, 14, 19, 22, 22, 12, 0
OFFSET
0,13
COMMENTS
A(n,k) is the number of partitions of n in which no parts are multiples of k.
A(n,k) is also the number of partitions of n into at most k-1 copies of each part.
FORMULA
G.f. of column k: Product_{j>=1} (1 - x^(k*j))/(1 - x^j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, ...
0, 2, 2, 3, 3, 3, ...
0, 2, 4, 4, 5, 5, ...
0, 3, 5, 6, 6, 7, ...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k*i*(i+1)/2<n, 0,
add((l->[0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
end:
A:= (n, k)-> b(n$2, k-1)[1]:
seq(seq(A(n, 1+d-n), n=0..d), d=0..16); # Alois P. Heinz, Oct 17 2018
MATHEMATICA
Table[Function[k, SeriesCoefficient[Product[(1 - x^(i k))/(1 - x^i), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[x^k, x^k]/QPochhammer[x, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
CROSSREFS
Main diagonal gives A000041.
Mirror of A061198.
Sequence in context: A189463 A287451 A113414 * A283308 A339959 A255636
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, May 11 2017
STATUS
approved