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%I #8 Sep 18 2022 12:37:49
%S 0,0,0,0,1,0,0,2,2,0,0,3,8,3,0,0,4,1,1,4,0,0,5,32,15,32,5,0,0,6,14,6,
%T 6,14,6,0,0,7,4,7,256,7,4,7,0,0,8,5,9,2,2,9,5,8,0,0,9,128,63,48,35,48,
%U 63,128,9,0,0,10,6,2,1,1,1,1,2,6,10,0,0,11,56,27,2048,47,60,47,2048,27,56,11,0
%N Square array, A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) * f(k)) where f(m) = A002487(m)/A002487(m+1) and g is the inverse of f.
%C The function f is a bijection from the nonnegative integers to the nonnegative rational numbers.
%C The positive integers, together with (x,y) -> A(x,y), form an abelian group isomorph to the multiplicative group of positive rational numbers (f and g act as isomorphisms).
%C Each row (or column), except the first, is a permutation of the nonnegative integers.
%H Rémy Sigrist, <a href="/A357144/a357144.gp.txt">PARI program</a>
%F A(n, k) = A(k, n).
%F A(n, 0) = 0.
%F A(n, 1) = n.
%F A(n, A054429(n)) = 1 for any n > 0.
%F A(m, A(n, k)) = A(A(m, n), k).
%F A(n, A(n-1, ... A(2, 1) ... )) = 2^(A002487(n+1)-1).
%e Array A(n, k) begins:
%e n\k | 0 1 2 3 4 5 6 7 8 9 10 11 12
%e ----+------------------------------------------------------------------
%e 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0
%e 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12
%e 2 | 0 2 8 1 32 14 4 5 128 6 56 17 16
%e 3 | 0 3 1 15 6 7 9 63 2 27 33 31 30
%e 4 | 0 4 32 6 256 2 48 1 2048 60 16 62 384
%e 5 | 0 5 14 7 2 35 1 47 20 3 1022 119 10
%e 6 | 0 6 4 9 48 1 60 3 32 510 12 13 72
%e 7 | 0 7 5 63 1 47 3 511 14 15 61 383 33
%e 8 | 0 8 128 2 2048 20 32 14 32768 4 320 26 512
%e 9 | 0 9 6 27 60 3 510 15 4 93 30 39 258
%e 10 | 0 10 56 33 16 1022 12 61 320 30 196 5 1008
%e 11 | 0 11 17 31 62 119 13 383 26 39 5 575 1
%e 12 | 0 12 16 30 384 10 72 33 512 258 1008 1 960
%o (PARI) See Links section.
%Y Cf. A002487, A054429, A354522, A355090.
%K nonn,tabl
%O 0,8
%A _Rémy Sigrist_, Sep 15 2022
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