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A028262 Elements in 3-Pascal triangle (by row).

%I A028262 #36 Jul 02 2025 16:01:55
%S A028262 1,1,1,1,3,1,1,4,4,1,1,5,8,5,1,1,6,13,13,6,1,1,7,19,26,19,7,1,1,8,26,
%T A028262 45,45,26,8,1,1,9,34,71,90,71,34,9,1,1,10,43,105,161,161,105,43,10,1,
%U A028262 1,11,53,148,266,322,266,148,53,11,1,1,12,64,201,414,588,588,414,201,64,12,1
%N A028262 Elements in 3-Pascal triangle (by row).
%H A028262 Reinhard Zumkeller, <a href="/A028262/b028262.txt">Rows n = 0..150 of triangle, flattened</a>
%H A028262 László Németh, <a href="http://math.colgate.edu/~integers/t41/t41.Abstract.html">Tetrahedron trinomial coefficient transform</a>, Integers (2019) Vol. 19, Article A41.
%H A028262 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A028262 After the 3rd row, use Pascal's rule.
%F A028262 From _Ralf Stephan_, Jan 31 2005: (Start)
%F A028262 T(n, k) = C(n, k) + C(n-2, k-1).
%F A028262 G.f.: (1 + x^2*y)/(1 - x*(1+y)). (End)
%F A028262 T(n+2,k+1) = A007318(n,k) - A007318(n+2,k+1); 0 < k < n. - _Reinhard Zumkeller_, Aug 02 2012
%F A028262 Sum_{k=0..n} T(n,k) = (n+1)*[n<2] + 5*2^(n-2)*[n>=2]. - _G. C. Greubel_, Apr 28 2021
%e A028262 Triangle begins:
%e A028262   1;
%e A028262   1 1;
%e A028262   1 3 1;
%e A028262   1 4 4 1;
%e A028262   1 5 8 5 1;
%e A028262   ...
%t A028262 T[n_, k_]:= If[n==1, 1, Binomial[n, k] + Binomial[n-2, k-1]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* _Jean-François Alcover_, Jan 28 2015 *)
%o A028262 (Haskell)
%o A028262 a028262 n k = a028262_tabl !! n !! k
%o A028262 a028262_row n = a028262_tabl !! n
%o A028262 a028262_tabl = [1] : [1,1] : iterate
%o A028262    (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [1,3,1]
%o A028262 -- _Reinhard Zumkeller_, Aug 02 2012
%o A028262 (Magma)
%o A028262 T:= func< n,k | n lt 2 select 1 else Binomial(n, k) + Binomial(n-2, k-1) >;
%o A028262 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 28 2021
%o A028262 (Sage)
%o A028262 def T(n,k): return 1 if n<2 else binomial(n,k) + binomial(n-2,k-1)
%o A028262 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 28 2021
%Y A028262 Cf. A007318, A028275, A072405.
%K A028262 nonn,nice,tabl
%O A028262 0,5
%A A028262 _Mohammad K. Azarian_
%E A028262 More terms from _James Sellers_