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A105728 Triangle read by rows: T(n,1) = 1, T(n,n) = n and for 1 < k < n: T(n,k) = T(n-1,k-1) + 2*T(n-1,k).

%I A105728 #23 Sep 08 2022 08:45:17
%S A105728 1,1,2,1,5,3,1,11,11,4,1,23,33,19,5,1,47,89,71,29,6,1,95,225,231,129,
%T A105728 41,7,1,191,545,687,489,211,55,8,1,383,1281,1919,1665,911,321,71,9,1,
%U A105728 767,2945,5119,5249,3487,1553,463,89,10,1,1535,6657,13183,15617,12223,6593,2479,641,109,11
%N A105728 Triangle read by rows: T(n,1) = 1, T(n,n) = n and for 1 < k < n: T(n,k) = T(n-1,k-1) + 2*T(n-1,k).
%C A105728 Sum of n-th row = 3^(n-1): Sum_{k=1..n} T(n,k) = A000244(n-1);
%C A105728 for n>1: T(n,2) = A083329(n-1), T(n,n-1) = A028387(n-2).
%H A105728 Reinhard Zumkeller, <a href="/A105728/b105728.txt">Rows n = 1..120 of triangle, flattened</a>
%e A105728 Triangle begins as:
%e A105728   1;
%e A105728   1,  2;
%e A105728   1,  5,  3;
%e A105728   1, 11, 11,  4;
%e A105728   1, 23, 33, 19,  5;
%e A105728   1, 47, 89, 71, 29, 6;
%e A105728 ...
%p A105728 T:= proc(n, k) option remember;
%p A105728       if k=1 then 1
%p A105728     elif k=n then n
%p A105728     else T(n-1, k-1) + 2*T(n-1, k)
%p A105728       fi
%p A105728     end:
%p A105728 seq(seq(T(n, k), k=1..n), n=1..12); # _G. C. Greubel_, Nov 13 2019
%t A105728 T[n_, k_]:= T[n, k]= If[k==1, 1, If[k==n, n, T[n-1, k-1] + 2*T[n-1, k]]];
%t A105728 Table[T[n, k], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Nov 13 2019 *)
%o A105728 (Haskell)
%o A105728 a105728 n k = a105728_tabl !! (n-1) !! (k-1)
%o A105728 a105728_row n = a105728_tabl !! (n-1)
%o A105728 a105728_tabl = iterate (\row -> zipWith (+) ([0] ++ tail row ++ [1]) $
%o A105728                                 zipWith (+) ([0] ++ row) (row ++ [0])) [1]
%o A105728 -- _Reinhard Zumkeller_, Jul 22 2013
%o A105728 (Magma)
%o A105728 function T(n,k)
%o A105728   if k eq 1 then return 1;
%o A105728   elif k eq n then return n;
%o A105728   else return T(n-1,k-1) + 2*T(n-1,k);
%o A105728   end if;
%o A105728   return T;
%o A105728 end function;
%o A105728 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 13 2019
%o A105728 (Sage)
%o A105728 @CachedFunction
%o A105728 def T(n, k):
%o A105728     if (k==1): return 1
%o A105728     elif (k==n): return n
%o A105728     else: return T(n-1,k-1) + 2*T(n-1, k)
%o A105728 [[T(n, k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Nov 13 2019
%Y A105728 Cf. A013609, A115068.
%K A105728 nonn,tabl
%O A105728 1,3
%A A105728 _Reinhard Zumkeller_, Apr 18 2005