Original entry on oeis.org
1, -1, -3, -20, -54, 4935, 403432, 23308238, -2635805834, -2939783620152, -1713742918458426, 602896713529233651, 9901041507182530035347, 52279007840299710266340246, -71905380320280305597098525356, -17521448585729172053338909789657052
Offset: 0
-
def s(k, n)
s = 0
(1..n).each{|i| s += (-1) ** (n / i + 1) * i ** k if n % i == 0}
s
end
def A(k, n)
ary = [1]
a = [0] + (1..n).map{|i| s(k + 1, i)}
(1..n).each{|i| ary << (1..i).inject(0){|s, j| s - a[j] * ary[-j]} / i}
ary
end
def A281268(n)
(0..n).map{|i| A(i, i)[-1]}
end
A284992
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)^(j^k) in powers of x.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 8, 3, 1, 1, 16, 35, 31, 16, 4, 1, 1, 32, 97, 119, 83, 28, 5, 1, 1, 64, 275, 457, 433, 201, 49, 6, 1, 1, 128, 793, 1763, 2297, 1476, 487, 83, 8, 1, 1, 256, 2315, 6841, 12421, 11113, 4962, 1141, 142, 10, 1, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 4, 8, 16, 32, 64, 128, ...
2, 5, 13, 35, 97, 275, 793, 2315, ...
2, 8, 31, 119, 457, 1763, 6841, 26699, ...
3, 16, 83, 433, 2297, 12421, 68393, 382573, ...
4, 28, 201, 1476, 11113, 85808, 678101, 5466916, ...
5, 49, 487, 4962, 52049, 561074, 6189117, 69540142, ...
-
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1, k)*binomial(i^k, j), j=0..n/i)))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14); # Alois P. Heinz, Oct 16 2017
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
Sum[b[n - i*j, i - 1, k]*Binomial[i^k, j], {j, 0, n/i}]]];
A[n_, k_] := b[n, n, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 10 2021, after Alois P. Heinz *)
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