A028326 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).

2, 2, 2, 2, 4, 2, 2, 6, 6, 2, 2, 8, 12, 8, 2, 2, 10, 20, 20, 10, 2, 2, 12, 30, 40, 30, 12, 2, 2, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 140, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2, 2, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2, 2, 22, 110, 330, 660, 924, 924, 660, 330, 110, 22, 2

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• tabl
• easy
• nice

Comments

Also number of binary vectors of length n+1 with k+1 runs (1 <= k <= n).

If the last two entries in each row are removed and 0 replaces the entries in a checkerboard pattern, we obtain

2;

0, 6;

2, 0, 12;

0, 10, 0, 20;

2, 0, 30, 0, 30;

0, 14, 0, 70, 0, 42;

2, 0, 56, 0, 140, 0, 56;

0, 18, 0, 168, 0, 252, 0, 72;

...

This plays the same role of recurrence coefficients for second differences of polynomials as triangle A074909 plays for the first differences. - R. J. Mathar, Jul 03 2013

From Roger Ford, Jul 06 2023: (Start)

T(n,k) = the number of closed meanders with n top arches, n+1 exterior arches and with k = the number of arches of length 1 - (n+1).

Example of closed meanders with 4 top arches and 5 exterior arches:

exterior arches are top arches or bottom arches without a covering arch

/\ = top arch length 1, \/ = bottom arch length 1

/ \ Top: /\=3 / \ / \ Top: /\=2

/\ / /\ \ /\ / /\ \ / /\ \

\ \/ / \ \/ / Bottom: \/=2 \/ \ \/ / \/ Bottom: /\=3

\/ \/ k=5-5=0 \/ k=5-5=0 T(4,0) = 2

____

/ \ Top: /\=3 / \ Top: /\=3

/\ / /\ /\ \ / /\ \ /\ /\

\ \/ / \/ \/ Bottom: \/=3 \/ \ \/ \/ / Bottom: \/=3

\/ k=6-5=1 \____/ k=6-5=1

____

/ \ Top: /\=3 / \ Top: /\=3

/ /\ /\ \ /\ /\ /\ / /\ \

\/ \/ \ \/ / Bottom: \/=3 \ \/ \/ / \/ Bottom: \/=3

\/ k=6-5=1 \____/ k=6-5=1 T(4,1) = 4

________

/ \ Top: /\=3

/ /\ /\ /\ \ /\ /\ /\ /\ Top: /\=4

\/ \/ \/ \/ Bottom: \/=4 \ \/ \/ \/ / Bottom: ||=3

k=7-5=2 \________/ k=7-5=2 T(4,2) = 2.

(End)

Examples

			Triangle begins:
  2;
  2,  2;
  2,  4,   2;
  2,  6,   6,   2;
  2,  8,  12,   8,   2;
  2, 10,  20,  20,  10,    2;
  2, 12,  30,  40,  30,   12,    2;
  2, 14,  42,  70,  70,   42,   14,    2;
  2, 16,  56, 112, 140,  112,   56,   16,   2;
  2, 18,  72, 168, 252,  252,  168,   72,  18,   2;
  2, 20,  90, 240, 420,  504,  420,  240,  90,  20,   2;
  2, 22, 110, 330, 660,  924,  924,  660, 330, 110,  22,  2;
  2, 24, 132, 440, 990, 1584, 1848, 1584, 990, 440, 132, 24, 2;
		

References

• I. Goulden and D. Jackson, Combinatorial Enumeration, John Wiley and Sons, 1983, page 76.

Links

• Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
• R. J. Mathar, Paving rectangular regions with rectangular tiles: tatami and non-tatami tilings, arXiv:1311.6135 [math.CO], 2013, Table 48.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A028327, A028328, A028329, A028330, A028331, A028332, A124927, A134058.

Programs

Haskell

a028326 n k = a028326_tabl !! n !! k
a028326_row n = a028326_tabl !! n
a028326_tabl = iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [2]
-- Reinhard Zumkeller, Mar 12 2012

Magma

[2*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 27 2021

Maple

T := proc(n, k) if k=0 then 2 elif k>n then 0 else T(n-1, k)+T(n-1, k-1) fi end:
for n from 0 to 13 do seq(T(n, k), k=0..n) od; # Zerinvary Lajos, Dec 16 2006

Mathematica

Table[2*Binomial[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Robert G. Wilson v, Mar 05 2012 *)

PARI

T(n,k) = 2*binomial(n,k) \\ Charles R Greathouse IV, Feb 07 2017

Python

from sympy import binomial
def T(n, k):
    return 2*binomial(n, k)
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 29 2017

Sage

flatten([[2*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 27 2021

Formula

G.f. for the number of length n binary words with k runs: (1-x+x*y)/(1-x-x*y) [Goulden and Jackson]. - Geoffrey Critzer, Mar 04 2012

Extensions

More terms from Donald Manchester, Jr. (s1199170(AT)cedarnet.cedarville.edu)