A118416 Triangle read by rows: T(n,k) = (2*k-1)*2^(n-1), 0 < k <= n.

1, 2, 6, 4, 12, 20, 8, 24, 40, 56, 16, 48, 80, 112, 144, 32, 96, 160, 224, 288, 352, 64, 192, 320, 448, 576, 704, 832, 128, 384, 640, 896, 1152, 1408, 1664, 1920, 256, 768, 1280, 1792, 2304, 2816, 3328, 3840, 4352, 512, 1536, 2560, 3584, 4608, 5632, 6656, 7680

Offset: 1

Author

Reinhard Zumkeller, Apr 27 2006

Keywords

• nonn
• tabl

Comments

Row sums give A014477: Sum_{k=1..n} T(n,k) = A014477(n-1);

central terms give A118415; T(2*k-1,k) = A058962(k-1);

T(n,1) = A000079(n-1);

T(n,2) = A007283(n-1) for n > 1;

T(n,3) = A020714(n-1) for n > 2;

T(n,4) = A005009(n-1) for n > 3;

T(n,5) = A005010(n-1) for n > 4;

T(n,n-1) = A118417(n-1) for n > 1;

T(n,n) = A014480(n-1) = A118413(n,n);

A001511(T(n,k)) = A002024(n,k);

A003602(T(n,k)) = A002260(n,k).

The alternating row sums, Sum_{k=1..n} (-1)^(k+1)*T(n,k), are: (a) in odd rows, the central term, T(n,(n+1)/2) = A058962((n-1)/2); (b) in even rows, the negation of the average of the two central terms, -(T(2n,n) + T(2n,+1))/2 = -A018215(m/2). The absolute values of the alternating row sums give the plain row means, Sum_{k=1..n} T(n,k)/n; the alternating sign row means are (-2)^(n-1). - Gregory Gerard Wojnar, Feb 10 2024

Examples

			Triangle begins:
   1;
   2,   6;
   4,  12,  20;
   8,  24,  40,  56;
  16,  48,  80, 112, 144;
  32,  96, 160, 224, 288, 352;
  64, 192, 320, 448, 576, 704, 832;
		

Links

• Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A118413, A117303, A054582.

Programs

Haskell

a118416 n k = a118416_tabl !! (n-1) !! (k-1)
a118416_row 1 = [1]
a118416_row n = (map (* 2) $ a118416_row (n-1)) ++ [a014480 (n-1)]
a118416_tabl = map a118416_row [1..]
-- Reinhard Zumkeller, Jan 22 2012

Maple

A118416 := proc(n,k) 2^(n-1)*(2*k-1) ; end proc: # R. J. Mathar, Sep 04 2011

Mathematica

Flatten[Table[(2k-1)2^(n-1),{n,10},{k,n}]] (* Harvey P. Dale, Aug 26 2014 *)

Python

from math import isqrt
def A118416(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(1-a)+(n<<1)-1<Chai Wah Wu, Jun 20 2025

Formula

T(n,k) = 2*T(n-1,k), 1 <= k < n; T(n,n) = A014480(n-1).