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A065941 Triangle T(n,k) = binomial(n-floor((k+1)/2), floor(k/2)). 64
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 3, 3, 1, 1, 1, 5, 4, 6, 3, 1, 1, 1, 6, 5, 10, 6, 4, 1, 1, 1, 7, 6, 15, 10, 10, 4, 1, 1, 1, 8, 7, 21, 15, 20, 10, 5, 1, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1, 1, 1, 10, 9, 36, 28, 56, 35, 35, 15, 6, 1, 1, 1, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Row sums give the Fibonacci sequence. So do the alternating row sums.

Triangle of coefficients of polynomials defined by p(0,x) = p(1,x) = 1, p(n+2,x) = x*p(n+1,x) + p(n,x). - Benoit Cloitre, May 08 2005

Another version of triangle in A103631. - Philippe Deléham, Jan 01 2009

The T(n,k) coefficients appear in appendix 2 of Parks' remarkable article "A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov" if we assume that the b(n) coefficients are all equal to 1 and ignore the first column. The complete version of this triangle including the first column is A103631. - Johannes W. Meijer, Aug 11 2011

From Gary W. Adamson, Sep 06 2011: (Start) Signed ++--++..., the roots are chaotic using f(x) = x^2 - 2 with cycle lengths shown in A003558 by n-th rows. Example: given row 3, x^3 + x^2 - 2x -1; the roots are (a = 1.24697,...; b = -.445041,...; c = -1.802937,...). Then (say using seed b with x^2 - 2) we obtain the trajectory -.445041,...-> -1.80193,...-> 1.24697,...; matching the entry "3" in A003558(3). Received from Herb Conn, Jan 2004:

Let x = 2*Cos 2A (A = Angle); then

Sin A/Sin A... = 1

Sin 3A/Sin A . = x + 1

Sin 5A/Sin A . = x^2 + x - 1

Sin 7A/Sin A . = x^3 + x - 2x - 1

Sin 9A/Sin A . = x^4 + x^3 - 3x^2 - 2x + 1

... (signed ++--++...). (End)

Or Pascal's triangle (A007318) with duplicated diagonals. Also triangle of coefficients of polynomials defined by P_0(x) = 1 and for n>=1, P_n(x) = F_n(x) + F_(n+1)(x), where F_n(x) is Fibonacci polynomial (cf. A049310): F_n(x) = sum{i=0,...,floor((n-1)/2)}C(n-i-1,i)*x^(n-2*i-1).- Vladimir Shevelev, Apr 12 2012

The matrix inverse is given by

1;

1,1;

0,-1,1;

0,1,-2,1;

0,0,1,-2,1;

0,0,-1,3,-3,1;

0,0,0,-1,3,-3,1;

0,0,0,1,-4,6,-4,1;

0,0,0,0,1,-4,6,-4,1;

... apart from signs the same as A124645. - R. J. Mathar, Mar 12 2013

REFERENCES

A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979. See Table 4.

Jay Kappraff, "Beyond Measure, A Guided Tour Through Nature, Myth and Number", World Scientific, 2002; p. 490.

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)

LINKS

Nathaniel Johnston, Rows 0..100, flattened

Henry W. Gould, A Variant of Pascal's Triangle , The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, p. 257-271, with corrections.

A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979. [Annotated scanned copy]

Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Journal of Dynamical Systems and Geometric Theories, Vol 2 (2004), p 65.

E. Munarini and N. Z. Salvi, Binary strings without zigzags, Séminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.

P.C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov , Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) p. 694-702.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

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

FORMULA

T(n,k) = binomial(n-floor((k+1)/2), floor(k/2)).

As a square array read by antidiagonals, this is given by T1(n, k) = binomial(floor(n/2) + k, k). - Paul Barry, Mar 11 2003

Triangle is a reflection of that in A066170 (absolute values). - Gary W. Adamson, Feb 16 2004

Recurrences: T(k, 0) = 1, T(k, n) = T(k-1, n) + T(k-2, n-2), or T(k, n) = T(k-1, n) + T(k-1, n-1) if n even, T(k-1, n-1) if n odd. - Ralf Stephan, May 17 2004

G.f.: sum[n, sum[k, T(k, n)x^ky^n]] = (1+xy)/(1-y-x^2y^2). sum[n>=0, T(k, n)y^n] = y^k/(1-y)^[k/2]. - Ralf Stephan, May 17 2004

T(n,k) = A108299(n,k)*A087960(k) = abs(A108299(n,k)). - Reinhard Zumkeller, Jun 01 2005

From Johannes W. Meijer, Aug 11 2011: (Start)

  T(n,k) = A046854(n,n-k) = abs(A066170(n,n-k)).

  T(n+k,n-k) = A109223(n,k).

  T(n,k) = sum(T(j, k-2), j=k-2..n-2), 2 <= k <= n, n>=2; T(n,0) =1, T(n+1,1) = 1, n>=0. (End)

For n > 1: T(n, k) = T(n-2, k) + T(n-1, k), 1 < k < n. - Reinhard Zumkeller, Apr 24 2013

EXAMPLE

Triangle begins:

1

1 1

1 1 1

1 1 2 1

1 1 3 2 1

1 1 4 3 3 1

...

MAPLE

A065941 := proc(n, k): binomial(n-floor((k+1)/2), floor(k/2)) end: seq(seq(A065941(n, k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 11 2011

A065941 := proc(n, k) option remember: local j: if k=0 then 1 elif k=1 then 1: elif k>=2 then add(procname(j, k-2), j=k-2..n-2) fi: end: seq(seq(A065941(n, k), k=0..n), n=0..11);  # Johannes W. Meijer, Aug 11 2011

MATHEMATICA

Flatten[Table[Binomial[n-Floor[(k+1)/2], Floor[k/2]], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Dec 11 2011 *)

PROG

(Haskell)

a065941 n k = a065941_tabl !! n !! k

a065941_row n = a065941_tabl !! n

a065941_tabl = iterate (\row ->

   zipWith (+) ([0] ++ row) (zipWith (*) (row ++ [0]) a059841_list)) [1]

-- Reinhard Zumkeller, May 07 2012

(PARI) T065941(n, k) = binomial(n-(k+1)\2, k\2); \\ Michel Marcus, Apr 28 2014

CROSSREFS

Cf. A065942 (central stalk sequence), A000045 (row sums), A108299.

Reflected version of A046854.

Some triangle sums (see A180662): A000045 (Fi1), A016116 (Kn21), A000295 (Kn23), A094967 (Fi2), A000931 (Ca2), A001519 (Gi3), A000930 (Ze3).

Cf. A003558.

Cf. A182579, A059841.

Sequence in context: A136568 A152157 A039961 * A108299 A123320 A054123

Adjacent sequences:  A065938 A065939 A065940 * A065942 A065943 A065944

KEYWORD

nonn,tabl,easy

AUTHOR

Len Smiley, Nov 29 2001

STATUS

approved

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Last modified February 25 22:17 EST 2018. Contains 299662 sequences. (Running on oeis4.)