A327917 Triangle T read by rows: T(k, n) = A(k-n, k) with the array A(k, n) = F(2*k+n) = A000045(2*k+n), for k >= 0 and n >= 0.

0, 1, 1, 3, 2, 1, 8, 5, 3, 2, 21, 13, 8, 5, 3, 55, 34, 21, 13, 8, 5, 144, 89, 55, 34, 21, 13, 8, 377, 233, 144, 89, 55, 34, 21, 13, 987, 610, 377, 233, 144, 89, 55, 34, 21, 2584, 1597, 987, 610, 377, 233, 144, 89, 55, 34, 6765, 4181, 2584, 1597, 987, 610, 377, 233, 144, 89, 55

Offset

0,4

Comments

This is the row reversed triangle A199334.

This is the analog of the array and the triangle A327916, where the positive odd numbers instead of the Fibonacci numbers are used.

The array A arises from the following Pascal-type triangles PF(k), for k >= 0, based on A000045 (Fibonacci). For example, the Pascal type triangle PF(k), for k = 3 is

0 1 1 2

1 2 3

3 5

8

For k -> infinity the left-aligned row sequences build the array A(k, n), with k >= 0 and n >= 0, that is, A(k, n) = F(2*k + n). See the example section for the first rows of A.

The first column sequence of A, {F(2*n) = A001906(n)}_{n>=0}, is the binomial transform of the first (k=0) row sequence of A, {F(n)}_{n>=0}.

The triangle T is the array A read by upwards antidiagonals.

Links

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

Formula

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

T(k, n) = A(k - n, n) = F(2*k - n), for k >= 0 and n = 0..k, with the Fibonacci numbers F = A000045.

Recurrence: T(k,0) = F(2*k), k >= 0, T(k, n) = T(k, n-1) - T(k-1, n-1), k >= 1, n = 1..k, and T(k, n) = 0 if k < n.

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

G(x, z) = Sum_{n=0} R(n, x)*z^n = z*(1 + x - 2*x*z)/((1 - 3*z + z^2)*(1 - x*z - (x*z)^2)).

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

Example

The Array A(k, n) begins:
k\n  0  1   2   3   4   5 ...
-----------------------------
0:   0  1   1   2   3   5 ...   F(n)
1:   1  2   3   5   8  13 ...   F(n+2)
2:   3  5   8  13  21  34 ...   F(n+4)
3:   8 13  21  34  55  89 ...   F(n+6)
4:  21 34  55  89 144 233 ...   F(n+8)
5:  55 89 144 233 377 610 ...   F(n+10)
...
---------------------------------------
The triangle T(k, n) begins:
k\n     0    1    2    3   4   5   6   7   8  9 10 ...
------------------------------------------------------
0:      0
1:      1    1
2:      3    2    1
3:      8    5    3    2
4:     21   13    8    5   3
5:     55   34   21   13   8   5
6:    144   89   55   34  21  13   8
7:    377  233  144   89  55  34  21  13
8:    987  610  377  233 144  89  55  34  21
9:   2584 1597  987  610 377 233 144  89  55 34
10:  6765 4181 2584 1597 987 610 377 233 144 89 55
...

Crossrefs

Cf. A000045, A199334 (row reversed), A001906, A327916.

Column sequences of T (no leading zeros) and A: from the shifted Fibonacci bisection {F(2*k) = A001906(k)} for even n, and {F(2*k+1) = A001519(k+1)}, for odd n.

Row sums: 2*A094292(n+1) = F(2*(n+1)) - F(n+1), n >= 0.

Alternating row sums: 2*A164267(n-1), n >= 0, with 0 for n = 0.

Sequence in context: A370527 A193924 A110439 * A065602 A237596 A292898

Adjacent sequences: A327914 A327915 A327916 * A327918 A327919 A327920

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 06 2019

Status

approved