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A105070 T(n,k)=2^k*binomial(n,2k+1) (0<=k<=floor((n-1)/2), n>=1). 6
1, 2, 3, 2, 4, 8, 5, 20, 4, 6, 40, 24, 7, 70, 84, 8, 8, 112, 224, 64, 9, 168, 504, 288, 16, 10, 240, 1008, 960, 160, 11, 330, 1848, 2640, 880, 32, 12, 440, 3168, 6336, 3520, 384, 13, 572, 5148, 13728, 11440, 2496, 64, 14, 728, 8008, 27456, 32032, 11648, 896, 15, 910 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row n contains ceil(n/2) terms. Row sums yield the Pell numbers (A000129). Column 1 yields A007290.

Eigenvector equals A118397, so that A118397(n) = Sum_{k=0..[n/2]} T(n+1,k)*A118397(k) for n>=0. - Paul D. Hanna, May 08 2006

Essentially a triangle, read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011

Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 07 2012

LINKS

Table of n, a(n) for n=1..58.

Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1.

J. Ivie, Problem B-161, Fibonacci Quarterly, 8 (1970), 107-108.

FORMULA

E.g.f.: exp(x)*sinh(x*sqrt(2*y))/sqrt(2*y), cf. A034867. - Vladeta Jovovic, Apr 06 2005

From Philippe Deléham, Apr 07 2012: (Start)

As DELTA-triangle T(n,k) with 0<=k<=n :

G.f.: (1-x+x^2-y*x^2)/(1-2*x+x^2-2*y*x^2).

T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(1,1) = T(2,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k<0 or if k>n. (End)

EXAMPLE

Triangle begins:

1;

2;

3,2;

4,8;

5,20,4;

6,40,24;

(2, -1/2, 1/2, 0, 0, ...) DELTA (0, 1, -1, 0, 0, ...) begins :

1

2, 0

3, 2, 0

4, 8, 0, 0

5, 20, 4, 0, 0

6, 40, 24, 0, 0, 0.

(1, 1, -1, 1, 0, 0, ...) DELTA (0, 0, 2, -2, 0, 0, ...) begins :

1

1, 0

2, 0, 0

3, 2, 0, 0

4, 8, 0, 0, 0

5, 20, 4, 0, 0, 0

6, 40, 24, 0, 0, 0, 0. - Philippe Deléham, Apr 07 2012

MAPLE

T:=(n, k)->binomial(n, 2*k+1)*2^k:for n from 1 to 15 do seq(T(n, k), k=0..floor((n-1)/2)) od; # yields sequence in triangular form

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x]

v[n_, x_] := u[n - 1, x] + v[n - 1, x]

Table[Factor[u[n, x]], {n, 1, z}]

Table[Factor[v[n, x]], {n, 1, z}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]  (* A207536 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]  (* A105070 *)

(* Clark Kimberling, Feb 18 2010 *)

CROSSREFS

Cf. A000129, A007290.

Cf. A118397 (eigenvector).

Sequence in context: A227961 A108838 A318176 * A154578 A059576 A274803

Adjacent sequences:  A105067 A105068 A105069 * A105071 A105072 A105073

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Apr 05 2005

STATUS

approved

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Last modified October 14 09:57 EDT 2019. Contains 327995 sequences. (Running on oeis4.)