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A028326 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k). 15
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

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

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

EXAMPLE

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

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 *)

PROG

(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

(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 xrange(21): print [T(n, k) for k in xrange(n + 1)] # Indranil Ghosh, Apr 29 2017

CROSSREFS

Cf. A028327, A007318.

Sequence in context: A237709 A250200 A097859 * A156046 A048003 A098219

Adjacent sequences:  A028323 A028324 A028325 * A028327 A028328 A028329

KEYWORD

nonn,tabl,easy,nice

AUTHOR

Mohammad K. Azarian

EXTENSIONS

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

STATUS

approved

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Last modified September 16 08:53 EDT 2019. Contains 327092 sequences. (Running on oeis4.)