A177696 Symmetrical triangle read by rows: T(n, k) = m*(T(n-1, k-1) + T(n-1, k)), where T(n, 1) = T(n, n) = n, and m = 2.

1, 2, 2, 3, 8, 3, 4, 22, 22, 4, 5, 52, 88, 52, 5, 6, 114, 280, 280, 114, 6, 7, 240, 788, 1120, 788, 240, 7, 8, 494, 2056, 3816, 3816, 2056, 494, 8, 9, 1004, 5100, 11744, 15264, 11744, 5100, 1004, 9, 10, 2026, 12208, 33688, 54016, 54016, 33688, 12208, 2026, 10

Offset: 1

Author

Roger L. Bagula, May 11 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
   1;
   2,    2;
   3,    8,     3;
   4,   22,    22,     4;
   5,   52,    88,    52,     5;
   6,  114,   280,   280,   114,     6;
   7,  240,   788,  1120,   788,   240,     7;
   8,  494,  2056,  3816,  3816,  2056,   494,     8;
   9, 1004,  5100, 11744, 15264, 11744,  5100,  1004,    9;
  10, 2026, 12208, 33688, 54016, 54016, 33688, 12208, 2026, 10;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A051597 (m=1).

Programs

Magma

function T(n, k) // T = A177696
  if k lt 1 or k gt n then return 0;
  elif k eq 1 or k eq n then return n;
  else return 2*(T(n-1, k-1) + T(n-1, k));
  end if;
end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 02 2024

Mathematica

m = 2;T[n_, k_]:= T[n, k]= If[k==1 || k==n, n, m*(T[n-1, k-1] + T[n-1, k])];Table[T[n, k], {n, 10}, {k, n}]//Flatten

SageMath

@CachedFunction
def T(n, k): # T = A177696
    if (k<0 or k>n): return 0
    elif (k==1 or k==n): return n
    else: return 2*(T(n-1, k-1) + T(n-1, k))
flatten([[T(n, k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 02 2024

Formula

T(n, k) = m*(T(n-1, k-1) + T(n-1, k)), where T(n, 1) = T(n, n) = n, and m = 2.

From G. C. Greubel, Oct 02 2024: (Start)

Sum_{k=1..n} T(n, k) = (1/9)*(7*4^n + 6*n + 2).

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1-(-1)^n)*(2-n) - [n=1]. (End)

Extensions

Edited by G. C. Greubel, Oct 02 2024