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Square array A(n,k) = P(A000010(k), (n+k-1)/k) if k divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.
4

%I #25 Dec 07 2019 12:18:29

%S 1,1,2,3,0,4,3,0,2,7,10,0,0,0,11,3,0,0,5,4,16,21,0,0,0,0,0,22,10,0,0,

%T 0,5,0,7,29,21,0,0,0,0,0,8,0,37,10,0,0,0,0,14,0,0,11,46,55,0,0,0,0,0,

%U 0,0,0,0,56,10,0,0,0,0,0,5,0,8,12,16,67,78,0,0,0,0,0,0,0,0,0,0,0,79,21,0,0,0,0,0,0,27,0,0,0,0,22,92,36,0,0,0,0,0,0,0,0,0,19,0,17,0,106

%N Square array A(n,k) = P(A000010(k), (n+k-1)/k) if k divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.

%C This is transpose of A286237, see comments there.

%H Antti Karttunen, <a href="/A286236/b286236.txt">Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array</a>

%F T(n,k) = A113998(n,k) * A286234(n,k).

%e The top left 12 X 12 corner of the array:

%e 1, 1, 3, 3, 10, 3, 21, 10, 21, 10, 55, 10

%e 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

%e 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

%e 7, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0

%e 11, 4, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0

%e 16, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0

%e 22, 7, 8, 0, 0, 5, 0, 0, 0, 0, 0, 0

%e 29, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0

%e 37, 11, 0, 8, 0, 0, 0, 14, 0, 0, 0, 0

%e 46, 0, 12, 0, 0, 0, 0, 0, 27, 0, 0, 0

%e 56, 16, 0, 0, 19, 0, 0, 0, 0, 14, 0, 0

%e 67, 0, 0, 0, 0, 0, 0, 0, 0, 0, 65, 0

%e The first 15 rows when viewed as a triangle:

%e 1,

%e 1, 2,

%e 3, 0, 4,

%e 3, 0, 2, 7,

%e 10, 0, 0, 0, 11,

%e 3, 0, 0, 5, 4, 16,

%e 21, 0, 0, 0, 0, 0, 22,

%e 10, 0, 0, 0, 5, 0, 7, 29,

%e 21, 0, 0, 0, 0, 0, 8, 0, 37,

%e 10, 0, 0, 0, 0, 14, 0, 0, 11, 46,

%e 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56,

%e 10, 0, 0, 0, 0, 0, 5, 0, 8, 12, 16, 67,

%e 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79,

%e 21, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 22, 92,

%e 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 17, 0, 106

%o (Scheme)

%o (define (A286236 n) (A286236bi (A002260 n) (A004736 n)))

%o (define (A286236bi row col) (if (not (zero? (modulo (+ row col -1) col))) 0 (let ((a (A000010 col)) (b (/ (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))

%o ;; Alternatively, with triangular indexing:

%o (define (A286236 n) (A286236tr (A002024 n) (A002260 n)))

%o (define (A286236tr n k) (A286236bi k (+ 1 (- n k))))

%o (Python)

%o from sympy import totient

%o def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2

%o def t(n, k): return 0 if n%k!=0 else T(totient(k), n/k)

%o for n in range(1, 21): print [t(n, k) for k in range(1, n + 1)][::-1] # _Indranil Ghosh_, May 10 2017

%Y Transpose: A286237.

%Y Cf. A000010, A000027, A113998, A286156, A286234, A286246.

%K nonn,tabl

%O 1,3

%A _Antti Karttunen_, May 05 2017