A327916 Triangle T(k, n) read by rows: Array A(k, n) = 2^k*(k + 1 + 2*n), k >= 0, n >= 0, read by antidiagonals upwards.

1, 4, 3, 12, 8, 5, 32, 20, 12, 7, 80, 48, 28, 16, 9, 192, 112, 64, 36, 20, 11, 448, 256, 144, 80, 44, 24, 13, 1024, 576, 320, 176, 96, 52, 28, 15, 2304, 1280, 704, 384, 208, 112, 60, 32, 17, 5120, 2816, 1536, 832, 448, 240, 128, 68, 36, 19, 11264, 6144, 3328, 1792, 960, 512, 272, 144, 76, 40, 21

Offset: 0

Author

Wolfdieter Lang, Oct 03 2019

Keywords

• nonn
• easy
• tabl

Comments

The array A(k, n) arises from the following Pascal-type triangles PTodd(k), k >= 0 based on the positive odd integers A005408.

For example, the Pascal-type triangle PTodd(k), for k = 3 is

1 3 5 7

4 8 12

12 20

32

Taken upside-down such triangles become so-called addition towers of height k+1 (Rechenturm in German elementary schools; thanks to my correspondent Bennet D.), starting with any k+1 numbers. Here the positive odd numbers are used.

The sequence s of the final number of these Pascal-type triangles PT(k), for k >= 0, begins 1, 4, 12, 32, ...; s(k) = (k+1)*2^k = A001787(k+1), for k >= 0.

For k -> infinity the left-aligned row sequences build the array A(k, n), with k >= 0 and n >= 0, namely A(k, n) = 2^k*(k + 2*n + 1); this array begins:

k\n 0 1 2 3 4 5 ...

-------------------------------

0: 1 3 5 7 9 11 ... {A005408(n)}

1: 4 8 12 16 20 24 ... {A008586(n+1)}

2: 12 20 28 36 44 52 ... {A017113(n+1)}

3: 32 48 64 80 96 112 ... {A008598(n+2)}

4: 80 112 144 176 208 240 ... {16*A005408(n+2)}

5: 192 256 320 384 448 512 ... {A152691(n+3)}

6: 448 576 704 832 960 1088 ... {64*A005408(n+3)}

...

The sequence s, the first (n=0) column of A, is always the binomial transform of the first (k=0) row in A.

A(k, n) = Sum_{j=0..k} binomial(k, j)*(2*(n+j)+1) = 2^k*(k + 1 + 2*n), for k >= 0 and n >= 0.

The corresponding antidiagonal-upwards read triangle is T(k, n) = A(k-n, n) = 2^(k-n)*(k + n + 1), n >= 0, k = 0..n.

If the nonnegative integers A001477 are used as k = 0 row of the array Anneg(k, n) = 2^(k-1)*(2*n + k), for k >= 0, n >= 0, with the triangle Tnneg(k, n) = Anneg(k-n, n) = (n + k)*2^(k-n-1), k >= 0, n = 0..k, then the s sequence is snneg(k) = Tnneg(k, 0) = k*2^{k-1} = A001787(k), the binomial transform of the sequence{A001477(n)}_{n>=0}. The triangle Tnneg begins [0], [1, 1], [4, 3, 2], [12, 8, 5, 3], [32, 20, 12, 7, 4], ... . See A062111 and the row-reversed triangle A152920 for other versions.

Examples

			The triangle T(k, n) begins:
   k\n    0    1    2    3   4   5   6   7  8  9 10 ...
  -----------------------------------------------------
   0:     1
   1:     4    3
   2:    12    8    5
   3:    32   20   12    7
   4:    80   48   28   16   9
   5:   192  112   64   36  20  11
   6:   448  256  144   80  44  24  13
   7:  1024  576  320  176  96  52  28  15
   8:  2304 1280  704  384 208 112  60  32 17
   9:  5120 2816 1536  832 448 240 128  68 36 19
  10: 11264 6144 3328 1792 960 512 272 144 76 40 21
  ...
		

Crossrefs

Column sequences without leading zeros are for n=0..9: A001787(n+1), A001792(n+1), A045623(n+2), A045891(n+3), A034007(n+4), A111297(n+3), A159694(n+1), A159695(n+1), A159696(n+1), A159697(n+1).

The sequence of (sub)diagonal k, for k >= 0, is the row k sequence of array A: {(k + 2*n + 1)*2^k}_{k >= 0}.

Row sums: A213569(k+1), k >= 0 (see the J. M. Bergot comments there).

Cf. A006211, A152920.

Programs

Mathematica

Table[2^#*(# + 1 + 2 n) &[k - n], {k, 0, 10}, {n, 0, k}] // Flatten (* Michael De Vlieger, Oct 03 2019 *)

Formula

Array A(k, n) = Sum_{j=0..k} binomial(k, j)*(2*(n+j) + 1) = 2^k*(k + 1+ 2*n), for k >= 0 and n >= 0.

Triangle T(k, n) = A(k-n, n) = 2^(k-n)*(k + n + 1), n >= 0, k = 0..n.

Recurrence: T(k, 0) = (k+1)*2^k = A001787(k+1), for k >= 0, and T(k, n) = T(k, n-1) - T(k-1, n-1), for n >= 1, k >= 1, with T(k, n) = 0 if k < n.

O.g.f. for row polynomials: G(z,x) = Sum_{n=0..k} R(k, x)*z^n =

(1 + x*z*(1 - 4*z))/((1 - 2*z)^2*(1 - x*z)^2).

T(k, 0) = Sum_{n=0..k} binomial(k,n)*T(n, n), k >= 0 (binomial transform).

Extensions

Definition corrected by Georg Fischer, Jul 13 2023