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%I #9 Sep 23 2016 23:02:04
%S 1,1,1,1,2,1,1,2,2,1,1,2,3,2,1,1,3,5,5,3,1,1,4,7,9,7,4,1,1,4,9,14,14,
%T 9,4,1,1,4,12,21,25,21,12,4,1,1,5,15,30,42,42,30,15,5,1,1,6,18,41,66,
%U 77,66,41,18,6,1
%N Triangle read by rows formed from golden-mean-like generalized factorials: t(n,m)=Round(phi*a(n)/(a(m)*a(n-m))), where phi=-(1 - Sqrt[5])/2, b(n)=b(n-1)+phi, a(n)=b(n)*a(n-1).
%C Row sums are {1, 2, 4, 6, 9, 18, 33, 56, 101, 186, 341, ...}.
%C The actual rational numbers are:
%C {1},
%C {1, 1},
%C {1, 3/2, 1},
%C {1, 2, 2, 1},
%C {1, 5/2, 10/3, 5/2, 1},
%C {1, 3, 5, 5, 3, 1},
%C {1, 7/2, 7, 35/4, 7, 7/2, 1},
%C {1, 4, 28/3, 14, 14, 28/3, 4, 1},
%C {1, 9/2, 12, 21, 126/5, 21, 12, 9/2, 1},
%C {1, 5, 15, 30, 42, 42, 30, 15, 5, 1},
%C {1, 11/2, 55/3, 165/4, 66, 77, 66, 165/4, 55/3, 11/2, 1}
%C ...
%F Phi=-(1 - Sqrt[5])/2; b(n)=b(n-1)+phi; a(n)=b(n)*a(n-1); t(n,m) = Round(Phi*a(n)/(a(m)*a(n-m))). - corrected by _Joshua Swanson_, Sep 16 2016
%F Empirically, t(n,m)=Round[(1/(m+1))*Binomial[n+1, m]]. - _Joshua Swanson_, Sep 16 2016
%e Triangle begins:
%e {1},
%e {1, 1},
%e {1, 2, 1},
%e {1, 2, 2, 1},
%e {1, 2, 3, 2, 1},
%e {1, 3, 5, 5, 3, 1},
%e {1, 4, 7, 9, 7, 4, 1},
%e {1, 4, 9, 14, 14, 9, 4, 1},
%e {1, 4, 12, 21, 25, 21, 12, 4, 1},
%e {1, 5, 15, 30, 42, 42, 30, 15, 5, 1},
%e {1, 6, 18, 41, 66, 77, 66, 41, 18, 6, 1}
%e ...
%t Clear[a, n, b, c] (* generalized Beta integer factorial of the golden mean*);
%t b[0] = -(1 - Sqrt[5])/2; b[n_] := b[n] = b[n - 1] - (1 - Sqrt[5])/2;
%t a[0] = -(1 - Sqrt[5])/2; a[n_] := a[n] = b[n]*a[n - 1]; (* combinations based on the gen-beta -factorials*)
%t c = Table[Table[FullSimplify[ExpandAll[((-1 + Sqrt[5])/2)*a[n]/(a[m]*a[n - m])]], {m, 0, n}], {n, 0, 10}];
%t Round[Flatten[c]]
%t (* edited by _Joshua Swanson_, Sep 16 2016 *)
%K nonn,tabl
%O 1,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 04 2008
%E Name corrected by _Joshua Swanson_, Sep 16 2016
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