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).

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, 41, 7, 1, 191, 545, 687, 489, 211, 55, 8, 1, 383, 1281, 1919, 1665, 911, 321, 71, 9, 1, 767, 2945, 5119, 5249, 3487, 1553, 463, 89, 10, 1, 1535, 6657, 13183, 15617, 12223, 6593, 2479, 641, 109, 11

Offset: 1

Author

Reinhard Zumkeller, Apr 18 2005

Keywords

• nonn
• tabl

Comments

Sum of n-th row = 3^(n-1): Sum_{k=1..n} T(n,k) = A000244(n-1);

for n>1: T(n,2) = A083329(n-1), T(n,n-1) = A028387(n-2).

Examples

			Triangle begins as:
  1;
  1,  2;
  1,  5,  3;
  1, 11, 11,  4;
  1, 23, 33, 19,  5;
  1, 47, 89, 71, 29, 6;
...
		

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Crossrefs

Cf. A013609, A115068.

Programs

Haskell

a105728 n k = a105728_tabl !! (n-1) !! (k-1)
a105728_row n = a105728_tabl !! (n-1)
a105728_tabl = iterate (\row -> zipWith (+) ([0] ++ tail row ++ [1]) $
                                zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Jul 22 2013

Magma

function T(n,k)
  if k eq 1 then return 1;
  elif k eq n then return n;
  else return T(n-1,k-1) + 2*T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 13 2019

Maple

T:= proc(n, k) option remember;
      if k=1 then 1
    elif k=n then n
    else T(n-1, k-1) + 2*T(n-1, k)
      fi
    end:
seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Nov 13 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==1, 1, If[k==n, n, T[n-1, k-1] + 2*T[n-1, k]]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Nov 13 2019 *)

Sage

@CachedFunction
def T(n, k):
    if (k==1): return 1
    elif (k==n): return n
    else: return T(n-1,k-1) + 2*T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 13 2019