login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A105809 A Fibonacci-Pascal matrix. 17
1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 5, 7, 7, 4, 1, 8, 12, 14, 11, 5, 1, 13, 20, 26, 25, 16, 6, 1, 21, 33, 46, 51, 41, 22, 7, 1, 34, 54, 79, 97, 92, 63, 29, 8, 1, 55, 88, 133, 176, 189, 155, 92, 37, 9, 1, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10, 1, 144, 232, 364, 530, 674, 709, 591 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums are A027934, antidiagonal sums are A010049(n+1). Inverse is A105810.

From Wolfdieter Lang, Oct 04 2014: (Start)

In the column k of this triangle (without leading zeros) is the k-fold iterated partial sums of the Fibonacci numbers, starting with 1. A000045(n+1), A000071(n+3), A001924(n+1), A014162(n+1), A014166(n+1),..., n >= 0. See the Riordan property. - Wolfdieter Lang, Oct 03 2014

For a combinatorial interpretation of these iterated partial sums see the H. Belbachir and A. Belkhir link. There table 1 shows in the rows these columns. In their notation (with r=k) f^(k)(n) = T(k,n+k).

The A-sequence of this Riordan triangle is [1, 1] (see the recurrence for T(n,k), k>=1, given in the formula section). The Z-sequence is A165326 = [1, repeat(1,-1)]. See the W. Lang link under A006232 for Riordan A- and Z-sequences.

The alternating row sums are A212804. (End)

LINKS

Reinhard Zumkeller, Rows n = 0..120 of table, flattened

H. Belbachir and A. Belkhir, Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers,Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.3.

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

Riordan array (1/(1-x-x^2), x/(1-x)); Number triangle T(n, k)=sum{j=0..n, binomial(n-j, k+j)}; T(n, 0)=A000045(n), T(n, m)=T(n-1, m-1)+T(n-1, m).

T(n,k)=sum{j=0..n, C(j,n+k-j)}. - Paul Barry, Oct 23 2006

G.f. of row polynomials sum(T(n,k)*x^k, k=0..n) is (1-z)/((1-z-z^2)*(1-(1+x)*z)) (Riordan property). - Wolfdieter Lang, Oct 04 2014

T(n, k) = binomial(n, k)*hypergeom([1, k/2-n/2, k/2-n/2+1/2],[k+1, -n],-4) for n>0. - Peter Luschny, Oct 10 2014

EXAMPLE

The triangle T(n,k) begins:

n\k   0   1   2    3    4    5    6    7    8   9  10 11 12 13 ...

0:    1

1:    1   1

2:    2   2   1

3:    3   4   3    1

4:    5   7   7    4    1

5:    8  12  14   11    5    1

6:   13  20  26   25   16    6    1

7:   21  33  46   51   41   22    7    1

8:   34  54  79   97   92   63   29    8    1

9:   55  88 133  176  189  155   92   37    9   1

10:  89 143 221  309  365  344  247  129   46  10   1

11: 144 232 364  530  674  709  591  376  175  56  11  1

12: 233 376 596  894 1204 1383 1300  967  551 231  67 12  1

13: 377 609 972 1490 2098 2587 2683 2267 1518 782 298 79 13  1

... reformatted and extended - Wolfdieter Lang, Oct 03 2014

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

Recurrence from Z-sequence (see a comment above): 8 = T(0,5) = (+1)*5 + (+1)*7 + (-1)*7 + (+1)*4 + (-1)*1 = 8. - Wolfdieter Lang, Oct 04 2014

MAPLE

T := (n, k) -> `if`(n=0, 1, binomial(n, k)*hypergeom([1, k/2-n/2, k/2-n/2+1/2], [k+1, -n], -4)); for n from 0 to 13 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Oct 10 2014

PROG

(Haskell)

a105809 n k = a105809_tabl !! n !! k

a105809_row n = a105809_tabl !! n

a105809_tabl = map fst $ iterate

   (\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [0]))) ([1], [1, 1])

-- Reinhard Zumkeller, Aug 15 2013

CROSSREFS

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A109906, A111006, A114197, A162741, A228074.

Cf.A165326 (Z-sequence), A027934 (row sums, see comment above), A212804 (alternating row sums). - Wolfdieter Lang, Oct 04 2014

Sequence in context: A209561 A283822 A210789 * A091594 A118032 A089692

Adjacent sequences:  A105806 A105807 A105808 * A105810 A105811 A105812

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, May 04 2005

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 20 02:47 EST 2018. Contains 299357 sequences. (Running on oeis4.)