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A327916 Triangle T(k, n) read by rows: Array A(k, n) = 2^k*(k + 1+ n), k >= 0, n >= 0, read by antidiagonals upwards. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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+ 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.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

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

EXAMPLE

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

...

MATHEMATICA

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

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)digonal 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. A06211, A152920.

Sequence in context: A240134 A193800 A061727 * A270025 A271199 A055527

Adjacent sequences:  A327913 A327914 A327915 * A327917 A327918 A327919

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Oct 03 2019

STATUS

approved

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Last modified November 15 01:25 EST 2019. Contains 329143 sequences. (Running on oeis4.)