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%I #18 Sep 10 2013 03:16:46
%S 1,1,1,1,2,1,1,3,3,1,1,4,10,4,1,1,5,41,41,5,1,1,6,206,1682,206,6,1,1,
%T 7,1237,346493,346493,1237,7,1,1,8,8660,428611842,120057399050,
%U 428611842,8660,8,1,1,9,69281,3711778551721,51458022952549550101,51458022952549550101,3711778551721,69281,9,1
%N Each term in the table is the product of the two terms above it + 1.
%C Row sums are A059731.
%H S. Kak, <a href="http://uk.arXiv.org/abs/physics/0411195">The Golden Mean and the Physics of Aesthetics</a>
%F a(m, n) = a(m-1, n-1)*a(m-1, n)+1, a(0, 0) = 1, a(m, n) = 0 iff n>m or n<0.
%e Triangle begins:
%e 1;
%e 1,1;
%e 1,2,1;
%e 1,3,3,1;
%e 1,4,10,4,1; ...
%p aaa := proc(m,n) option remember; if n>m or n<0 then 0; elif m=0 and n=0 then 1; else aaa(m-1,n-1)*aaa(m-1,n)+1; fi; end;
%t a[0, 0] = 1; a[m_, n_] /; (n > m || n < 0) = 0; a[m_, n_] := a[m, n] = a[m-1, n-1]*a[m-1, n] + 1; Table[a[m, n], {m, 0, 9}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Sep 10 2013 *)
%o (Haskell)
%o a059922 n k = a059922_tabl !! n !! k
%o a059922_flattened = concat a059922_tabl
%o a059922_tabl = iterate (\rs ->
%o zipWith (+) (0 : reverse (0 : replicate (length rs - 1) 1))
%o $ zipWith (*) ([1] ++ rs) (rs ++ [1])) [1]
%o a059730 n = a059922_tabl !! n !! (n-3)
%o a059731 n = sum (a059922_tabl !! n)
%o a059732 n = a059922_tabl !! (2*n) !! n
%o a059733 n = a059922_tabl !! n !! n `div` 2
%o -- _Reinhard Zumkeller_, Jun 22 2011
%Y Cf. A007318, A059730 - A059733.
%K easy,nice,nonn,tabl
%O 0,5
%A _Fabian Rothelius_, Feb 09 2001
%E More terms from _N. J. A. Sloane_ and Larry Reeves, Feb 09 2001.
%E Corrected by Jonathan Wellons (wellons(AT)gmail.com), May 24 2008
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