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
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
Programs
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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);
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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)