A105070 T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.

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, 12012, 51480, 80080, 43680, 6720, 128

Offset: 1

Author

Emeric Deutsch, Apr 05 2005

Keywords

• nonn
• tabf

Comments

Row n contains ceiling(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

Examples

			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
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened
• 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.

Crossrefs

Cf. A000129, A007290.

Cf. A118397 (eigenvector).

Programs

Magma

[2^k*Binomial(n,2*k+1): k in [0..Floor((n-1)/2)], n in [1..15]]; // G. C. Greubel, Mar 15 2020

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 *)
Table[2^k*Binomial[n, 2*k+1], {n, 15}, {k,0,Floor[(n-1)/2]}]//Flatten (* G. C. Greubel, Mar 15 2020 *)

Sage

[[2^k*binomial(n,2*k+1) for k in (0..floor((n-1)/2))] for n in (1..15)] # G. C. Greubel, Mar 15 2020

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)

Sum_{k=0..floor((n-1)/2)} T(n,k) = { P(n) (A000129(n)), A215928(n), (-1)^(n-1) *A077985(n-1), -A176981(n+1), (-1)^(n-1)*A215936(n+2) }, for n >= 1. - G. C. Greubel, Mar 15 2020