A327031 A(n, k) = Sum_{d|n} phi(d)*T(n/d, k) if n > 0 and A(0, k) = 0 where T(n, k) = binomial(n+k-1, n). Square array read by ascending antidiagonals, with n, k >= 0.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 8, 9, 4, 0, 0, 5, 12, 16, 14, 5, 0, 0, 6, 14, 27, 28, 20, 6, 0, 0, 7, 21, 33, 53, 45, 27, 7, 0, 0, 8, 20, 56, 72, 95, 68, 35, 8, 0, 0, 9, 28, 54, 132, 146, 159, 98, 44, 9, 0, 0, 10, 30, 84, 144, 285, 276, 252, 136, 54, 10, 0

Offset: 0

Author

Peter Luschny, Aug 25 2019

Keywords

• nonn
• tabl

Examples

			[0] 0, 0,  0,  0,   0,   0,    0,    0,     0,     0, ... A000004
[1] 0, 1,  2,  3,   4,   5,    6,    7,     8,     9, ... A001477
[2] 0, 2,  5,  9,  14,  20,   27,   35,    44,    54, ... A000096
[3] 0, 3,  8, 16,  28,  45,   68,   98,   136,   183, ... A255993 (conj.)
[4] 0, 4, 12, 27,  53,  95,  159,  252,   382,   558, ... A327032
[5] 0, 5, 14, 33,  72, 146,  276,  490,   824,  1323, ...
[6] 0, 6, 21, 56, 132, 285,  572, 1078,  1924,  3276, ...
[7] 0, 7, 20, 54, 144, 360,  828, 1758,  3480,  6489, ...
[8] 0, 8, 28, 84, 236, 615, 1479, 3297,  6869, 13491, ...
[9] 0, 9, 30, 93, 284, 815, 2150, 5215, 11728, 24694, ...
     A209295,
		

Crossrefs

Cf. A000004 (n=0), A001477 (n=1), A000096 (n=2), A255993 (n=3 conj.), A327032 (n=4), A209295, A097805.

Programs

Maple

DivisorSquareArray := proc(p, T, len) local row:
row := (n, k) -> add(p(d)*T(n/d, k), d = numtheory:-divisors(n)):
seq(lprint(seq(add(j, j=row(n, k)), k=0..len-1)), n=0..len-1) end:
DivisorSquareArray(numtheory:-phi, (n, k) -> binomial(n+k-1, n), 9);

SageMath

def DivisorSquareArray(p, T, Len):
    D = [[0]*Len]
    for n in (1..Len-1):
        r = lambda k: [p(d)*T(n//d, k) for d in divisors(n)]
        L = [sum(r(k)) for k in (0..Len-1)]
        D.append(L)
    return D
def T(n, k): return binomial(n + k - 1, n)
DivisorSquareArray(euler_phi, T, 10)