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A141604 Triangle, read by rows, T(n,k) = round(A006720(n)/(A006720(n-k)*A006720(k))).

%I A141604 #19 Sep 21 2024 08:41:36
%S A141604 1,1,1,1,1,1,1,1,1,1,1,2,2,2,1,1,2,3,3,2,1,1,2,4,7,4,2,1,1,3,8,12,12,
%T A141604 8,3,1,1,3,8,20,15,20,8,3,1,1,5,14,45,52,52,45,14,5,1,1,5,26,66,109,
%U A141604 170,109,66,26,5,1
%N A141604 Triangle, read by rows, T(n,k) = round(A006720(n)/(A006720(n-k)*A006720(k))).
%C A141604 The round-function in the definition is round-to-nearest, not Mathematica's round-to-even. - _R. J. Mathar_, Jul 12 2012
%H A141604 G. C. Greubel, <a href="/A141604/b141604.txt">Rows n = 0..50 of the triangle, flattened</a>
%e A141604 Triangle begins:
%e A141604   1;
%e A141604   1,   1;
%e A141604   1,   1,   1;
%e A141604   1,   1,   1,   1;
%e A141604   1,   2,   2,   2,   1;
%e A141604   1,   2,   3,   3,   2,   1;
%e A141604   1,   2,   4,   7,   4,   2,   1;
%e A141604   1,   3,   8,  12,  12,   8,   3,   1;
%e A141604   1,   3,   8,  20,  15,  20,   8,   3,   1;
%e A141604   1,   5,  14,  45,  52,  52,  45,  14,   5,   1;
%e A141604   1,   5,  26,  66, 109, 170, 109,  66,  26,   5,   1;
%p A141604 A141604 := proc(n,m)
%p A141604         round(A006720(n)/A006720(n-m)/A006720(m)) ;
%p A141604 end proc:
%p A141604 seq(seq(A141604(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jul 12 2012
%t A141604 f[n_]:= f[n]= If[n<4, 1, (f[n-1]*f[n-3] +f[n-2]^2)/f[n-4]]; (* A006720 *)
%t A141604 A141604[n_, k_]:= Round[f[n]/(f[k]*f[n-k])];
%t A141604 Table[A141604[n,k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Sep 21 2024 *)
%o A141604 (Magma)
%o A141604 A006720:= [n le 4 select 1 else (Self(n-1)*Self(n-3)+Self(n-2)^2)/Self(n-4): n in [1..30]];
%o A141604 A141604:= func< n,k | Round(A006720[n+1]/(A006720[k+1]*A006720[n-k+1])) >;
%o A141604 [A141604(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 21 2024
%o A141604 (SageMath)
%o A141604 def f(n): # f = A006720
%o A141604     if n<4: return 1
%o A141604     else: return (f(n-1)*f(n-3) +f(n-2)^2)/f(n-4)
%o A141604 def A141604(n,k): return round(f(n)/(f(k)*f(n-k)))
%o A141604 flatten([[A141604(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 21 2024
%Y A141604 Cf. A006720.
%K A141604 nonn,less,tabl
%O A141604 0,12
%A A141604 _Roger L. Bagula_ and _Gary W. Adamson_, Aug 21 2008