login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A338198 Triangle read by rows, T(n,k) = ((k+1)*2^(n-k)-(k-2)*(-1)^(n-k))/3 for 0 <= k <= n. 0
1, 0, 1, 2, 1, 1, 2, 3, 2, 1, 6, 5, 4, 3, 1, 10, 11, 8, 5, 4, 1, 22, 21, 16, 11, 6, 5, 1, 42, 43, 32, 21, 14, 7, 6, 1, 86, 85, 64, 43, 26, 17, 8, 7, 1, 170, 171, 128, 85, 54, 31, 20, 9, 8, 1, 342, 341, 256, 171, 106, 65, 36, 23, 10, 9, 1, 682, 683, 512, 341, 214, 127, 76, 41, 26, 11, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This triangle is related to the Jacobsthal numbers (A001045).

LINKS

Table of n, a(n) for n=0..77.

FORMULA

T(n,n) = 1 for n >= 0; T(n,n-1) = n-1 for n > 0.

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

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

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

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

Row sums are A083579(n+1) for n >= 0.

G.f. of column k >= 0: (1+(k-1)*t) * t^k / (1-t-2*t^2).

G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = (1 - (1+x)*t + 2*x*t^2) / ((1 - x*t)^2 * (1 - t - 2*t^2)).

Conjecture: Let M(n,k) be the matrix inverse of T(n,k), seen as a matrix. Then M(i,j) = 0 if j < 0 or j > i, M(n,n) = 1 for n >= 0, M(n,n-1) = 1-n for n > 0, and M(n,k) = (-1)^(n-k) * (k^2-2) * (n-2)! / k! for 0 <= k <= n-2.

EXAMPLE

The triangle T(n,k) for 0 <= k <= n starts:

n\k :    0     1     2    3    4    5    6   7   8   9

======================================================

  0 :    1

  1 :    0     1

  2 :    2     1     1

  3 :    2     3     2    1

  4 :    6     5     4    3    1

  5 :   10    11     8    5    4    1

  6 :   22    21    16   11    6    5    1

  7 :   42    43    32   21   14    7    6   1

  8 :   86    85    64   43   26   17    8   7   1

  9 :  170   171   128   85   54   31   20   9   8   1

etc.

MATHEMATICA

Table[((k + 1)*2^(n - k) - (k - 2)*(-1)^(n - k))/3, {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 15 2020 *)

CROSSREFS

For columns k = 0 to 8 see A078008, A001045, A000079, A001045, A084214, A014551, A083595, A083582, A259713 respectively.

Sequence in context: A201384 A238348 A143066 * A091598 A144021 A334591

Adjacent sequences:  A338195 A338196 A338197 * A338199 A338200 A338201

KEYWORD

nonn,easy,tabl

AUTHOR

Werner Schulte, Oct 15 2020

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 4 16:25 EDT 2021. Contains 346447 sequences. (Running on oeis4.)