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%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