|
| |
|
|
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
|
|
|
|
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 ...
-------------------------------
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.
|
|
|
LINKS
|
|
|
|
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).
|
|
|
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
|
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}.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
