OFFSET
0,6
COMMENTS
Conjecture: For p prime, A(n, p) == -1 (mod p) for n >= 0.
Conjecture: Let n >= 0, k >= 1 and k != 4. Then k divides A(n, k) if and only if k is not prime.
From Mélika Tebni, Jul 04 2022: (Start)
Conjecture: The polynomials of A355259 generate the k+1 column of this array.
Conjecture: For p prime and n even, (A(n, p) / (p - 1)) == 1 (mod p). (End)
FORMULA
A(n, k) = k!*Sum_{j=0..k-1} binomial(k + n - 1, k - j - 1) / (j + 1).
A(n, k) = k!*Sum_{j=1..k} binomial(n + k - j - 1, n - 1)*(2^j - 1) / j.
A(n, k) = k!*binomial(n + k - 1, k - 1)*hypergeom([1, 1, 1 - k], [2, n + 1], -1)) except for A(0, 0) = 0.
EXAMPLE
Table A(n, k) begins:
[0] 0, 1, 3, 14, 90, 744, 7560, 91440, 1285200, ... A029767
[1] 0, 1, 5, 29, 206, 1774, 18204, 218868, 3036144, ... A103213
[2] 0, 1, 7, 50, 406, 3804, 41028, 506064, 7084656, ... A355171
[3] 0, 1, 9, 77, 714, 7374, 85272, 1102968, 15908400, ... A355372
[4] 0, 1, 11, 110, 1154, 13144, 164136, 2251920, 33923760, ... A355407
[5] 0, 1, 13, 149, 1750, 21894, 295500, 4320420, 68487120, ... A355414
[6] 0, 1, 15, 194, 2526, 34524, 502644, 7838928, 131198544, ...
[7] 0, 1, 17, 245, 3506, 52054, 814968, 13543704, 239548176, ...
MAPLE
egf := n -> log((1 - x)/(1 - 2*x))/(1 - x)^n:
ser := n -> series(egf(n), x, 22):
row := n -> seq(k!*coeff(ser(n), x, k), k = 0..8):
seq(print(row(n)), n = 0..8);
# Alternative:
A := (n, k) -> add(k!*binomial(k + n - 1, k - j - 1)/(j + 1), j = 0..k-1):
seq(print(seq(A(n, k), k = 0..8)), n = 0..7);
MATHEMATICA
A[0, 0] = 0; A[n_, k_] := k! * Binomial[n+k-1, k - 1] * HypergeometricPFQ[{1 - k, 1, 1}, {2, n + 1}, -1];
Table[A[n, k], {n, 0, 8}, {k, 0, 8}] // TableForm
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny and Mélika Tebni, Jul 01 2022
STATUS
approved