A105851 Binomial transform triangle, read by rows.

1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 20, 12, 5, 1, 32, 48, 32, 16, 6, 1, 64, 112, 80, 44, 20, 7, 1, 128, 256, 192, 112, 56, 24, 8, 1, 256, 576, 448, 272, 144, 68, 28, 9, 1, 512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1, 1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1

Offset: 0

Author

Gary W. Adamson, Apr 23 2005

Keywords

• nonn
• tabl

Comments

Let P = Pascal's triangle as an infinite lower triangular matrix and A is the infinite array of arithmetic sequences as shown in A077028:

1, 1, 1, 1, 1, ...

1, 2, 3, 4, 5, ...

1, 3, 5, 7, 9, ...

1, 4, 7, 10, 13, ...

1, 5, 9, 13, 17, ...

Perform the operation P * A, getting a new array with each column being the binomial transform of an arithmetic sequence. Take antidiagonals of the new array, then by rows = the triangle of A105851.

Examples

			Column 3: 1, 5, 16, 44, 112, ... (A053220) is the binomial transform of 3k+1 (A016777: 1, 4, 7, ...).
Triangle begins:
     1;
     2,    1;
     4,    3,    1;
     8,    8,    4,    1;
    16,   20,   12,    5,   1;
    32,   48,   32,   16,   6,   1;
    64,  112,   80,   44,  20,   7,   1;
   128,  256,  192,  112,  56,  24,   8,  1;
   256,  576,  448,  272, 144,  68,  28,  9,  1;
   512, 1280, 1024,  640, 352, 176,  80, 32, 10,  1;
  1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1;
  ...
		

Crossrefs

Cf. A077028, A001792, A001787, A053220, A016777, A014480.

Programs

Magma

/* As triangle */ [[(2+k*(n-k))*2^(n-k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 26 2015

Maple

seq(seq((2 + k*(n - k))*2^(n-k-1),k=0..n),n=0..10); # Peter Bala, Jul 26 2015

Mathematica

t[n_, k_]:=(2 + k (n - k)) 2^(n - k - 1); Table[t[n - 1, k - 1], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jul 26 2015 *)

Formula

n-th column of the triangle is the binomial transform of the arithmetic sequence (n*k + 1), (k = 0, 1, 2, ...).

From Peter Bala, Jul 26 2015: (Start)

T(n,k) = (2 + k*(n - k))*2^(n-k-1) for 0 <= k <= n.

O.g.f.: (1 - x*(2 + t) + 3*t*x^2)/((1 - 2*x)^2*(1 - t*x)^2) = 1 + (2 + t)*x + (4 + 3*t + t^2)*x^2 + ....

k-th column g.f.: (1 + (k - 2)*x)/(1 - 2*x)^2. Cf. A077028. (End)

Extensions

More terms from Philippe Deléham, Mar 31 2007