cp's OEIS Frontend

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.

%I A327031 #10 Aug 27 2019 00:45:40
%S A327031 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,
%T A327031 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,
%U A327031 132,146,159,98,44,9,0,0,10,30,84,144,285,276,252,136,54,10,0
%N 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.
%e A327031 [0] 0, 0,  0,  0,   0,   0,    0,    0,     0,     0, ... A000004
%e A327031 [1] 0, 1,  2,  3,   4,   5,    6,    7,     8,     9, ... A001477
%e A327031 [2] 0, 2,  5,  9,  14,  20,   27,   35,    44,    54, ... A000096
%e A327031 [3] 0, 3,  8, 16,  28,  45,   68,   98,   136,   183, ... A255993 (conj.)
%e A327031 [4] 0, 4, 12, 27,  53,  95,  159,  252,   382,   558, ... A327032
%e A327031 [5] 0, 5, 14, 33,  72, 146,  276,  490,   824,  1323, ...
%e A327031 [6] 0, 6, 21, 56, 132, 285,  572, 1078,  1924,  3276, ...
%e A327031 [7] 0, 7, 20, 54, 144, 360,  828, 1758,  3480,  6489, ...
%e A327031 [8] 0, 8, 28, 84, 236, 615, 1479, 3297,  6869, 13491, ...
%e A327031 [9] 0, 9, 30, 93, 284, 815, 2150, 5215, 11728, 24694, ...
%e A327031      A209295,
%p A327031 DivisorSquareArray := proc(p, T, len) local row:
%p A327031 row := (n, k) -> add(p(d)*T(n/d, k), d = numtheory:-divisors(n)):
%p A327031 seq(lprint(seq(add(j, j=row(n, k)), k=0..len-1)), n=0..len-1) end:
%p A327031 DivisorSquareArray(numtheory:-phi, (n, k) -> binomial(n+k-1, n), 9);
%o A327031 (SageMath)
%o A327031 def DivisorSquareArray(p, T, Len):
%o A327031     D = [[0]*Len]
%o A327031     for n in (1..Len-1):
%o A327031         r = lambda k: [p(d)*T(n//d, k) for d in divisors(n)]
%o A327031         L = [sum(r(k)) for k in (0..Len-1)]
%o A327031         D.append(L)
%o A327031     return D
%o A327031 def T(n, k): return binomial(n + k - 1, n)
%o A327031 DivisorSquareArray(euler_phi, T, 10)
%Y A327031 Cf. A000004 (n=0), A001477 (n=1), A000096 (n=2), A255993 (n=3 conj.), A327032 (n=4), A209295, A097805.
%K A327031 nonn,tabl
%O A327031 0,8
%A A327031 _Peter Luschny_, Aug 25 2019