A028330 -id:A028330 - cp's OEIS frontend

A028329 Twice central binomial coefficients.

2, 4, 12, 40, 140, 504, 1848, 6864, 25740, 97240, 369512, 1410864, 5408312, 20801200, 80233200, 310235040, 1202160780, 4667212440, 18150270600, 70690527600, 275693057640, 1076515748880, 4208197927440, 16466861455200, 64495207366200, 252821212875504, 991837065896208

Offset: 0

Mohammad K. Azarian

Central elements in the even-Pascal triangle A028326.
If Y is a 3-subset of an 2n-set X then, for n>=3, a(n-1) is the number of (n+1)-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
a(n) denotes the number of ways one can reach the (n,n) point in an n X n grid via the point (n-1, n-1) starting from (0,0) when moving right and up is allowed [From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 29 2009]
It appears that a(n-1) is also the number of quivers in the mutation class of twisted types BD_n and CD_n for n >= 3. - Christian Stump, Nov 03 2010
This is the case m = n+1 in the Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - Bruno Berselli, Apr 27 2012
From Ran Pan, Feb 01 2016: (Start)
a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that bounce off the diagonal y = x an even number of times. Details can be found in Section 4.2 in Pan and Remmel's link.
a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that cross the diagonal y = x an even number of times. Details can be found in Section 4.3 in Pan and Remmel's link. (End)

Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third Edition), page 11.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Bisection of A047073, A063886.
First differences of A054113.
Cf. A000984, A095660, A100320, A162551.
Cf. A028326, A028327, A028328, A028330, A028331, A028332.

[2*(n+1)*Catalan(n): n in [0..30]]; // G. C. Greubel, Jul 13 2024

seq(add(binomial(2*n,n),k=1..2),n=0..23); # Zerinvary Lajos, Dec 14 2007

Table[2Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, Aug 08 2011 *)

PARI
```
a(n)=2*binomial(2*n,n)
```

[2*binomial(2*n,n) for n in range(31)] # G. C. Greubel, Jul 13 2024

G.f.: 2/sqrt(1 - 4*x).
a(n) = 2*A000984(n).
a(n) = 2 * binomial(2*n, n).
a(n) = A100320(n) = A095660(2*n,n) for n > 0. - Reinhard Zumkeller, Apr 08 2012
G.f.: G(0), where G(k)= 1 + 1/(1 - 2*x*(2*k + 1)/(2*x*(2*k + 1) + (k + 1)/ G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
a(n) = binomial(2*n+2, n+1) - A162551(n). - Ran Pan, Feb 01 2016
D-finite with recurrence: n*a(n) + 2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
E.g.f.: 2*exp(2*x)*BesselI(0, 2*x). - Stefano Spezia, May 11 2024

Edited by Michael Somos, Sep 13 2003

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

Original entry on oeis.org

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

Mohammad K. Azarian

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)

			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;

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

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

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

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

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

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

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

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

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

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

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

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

A028327 Elements in the even-Pascal triangle A028326 that are not 2.

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A028326, A028328, A028329, A028330, A028331, A028332.

[2*Binomial(n, k): k in [1..n-1], n in [1..12]]; // G. C. Greubel, Jul 13 2024

Table[2*Binomial[n,k], {n,13}, {k,n-1}]//Flatten (* G. C. Greubel, Jul 13 2024 *)

flatten([[2*binomial(n,k) for k in range(1,n)] for n in range(2,14)]) # G. C. Greubel, Jul 13 2024

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

A028328 Distinct elements in the even-Pascal triangle A028326.

Original entry on oeis.org

2, 4, 6, 8, 12, 10, 20, 30, 40, 14, 42, 70, 16, 56, 112, 140, 18, 72, 168, 252, 90, 240, 420, 504, 22, 110, 330, 660, 924, 24, 132, 440, 990, 1584, 1848, 26, 156, 572, 1430, 2574, 3432, 28, 182, 728, 2002, 4004, 6006, 6864, 210, 910, 2730, 10010, 12870, 32

Offset: 0

Mohammad K. Azarian

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A028326, A028327, A028329, A028330, A028331, A028332.

DeleteDuplicates[Table[2*Binomial[n,k], {n,0,30}, {k,0,n}]//Flatten] (* G. C. Greubel, Jul 13 2024 *)

A028326=flatten([[2*binomial(n,k) for k in range(n+1)] for n in range(31)])
def a(seq): # order preserving
    nd = [] # no duplicates
    [nd.append(i) for i in seq if not nd.count(i) and i%2==0]
    return nd
a(A028326) # A028328 # G. C. Greubel, Jul 13 2024

More terms from James Sellers, Dec 08 1999

A028331 Elements to the right of the central elements of the even-Pascal triangle A028326 that are not 2.

Original entry on oeis.org

6, 8, 20, 10, 30, 12, 70, 42, 14, 112, 56, 16, 252, 168, 72, 18, 420, 240, 90, 20, 924, 660, 330, 110, 22, 1584, 990, 440, 132, 24, 3432, 2574, 1430, 572, 156, 26, 6006, 4004, 2002, 728, 182, 28, 12870, 10010, 6006, 2730, 910, 210, 30, 22880, 16016

Offset: 0

Mohammad K. Azarian

			This sequence represents the following portion of A028330(n,k), with x being the elements of A028329(n):
  x;
  .,  .;
  .,  x,  .;
  .,  .,  6,  .;
  .,  .,  x,  8,  .;
  .,  .,  ., 20, 10,   .;
  .,  .,  .,  x, 30,  12,   .;
  .,  .,  .,  ., 70,  42,  14,    .;
  .,  .,  .,  .,  x, 112,  56,   16,   .;
  .,  .,  .,  .,  ., 252, 168,   72,  18,   .;
  .,  .,  .,  .,  .,   x, 420,  240,  90,  20,   .;
  .,  .,  .,  .,  .,   ., 924,  660, 330, 110,  22,  .;
  .,  .,  .,  .,  .,   .,   x, 1584, 990, 440, 132, 24, .;
As an irregular triangle:
    6;
    8;
   20,  10;
   30,  12;
   70,  42,  14;
  112,  56,  16;
  252, 168,  72,  18;
  420, 240,  90,  20;
  924, 660, 330, 110, 22;

G. C. Greubel, Rows n = 0..100 of the irregular triangle, flattened

Cf. A028326, A028327, A028328, A028329, A028330, A028332.

Cf. A272514, A286033.

[2*Binomial(n+3,k): k in [Floor((n+5)/2)..n+2], n in [0..12]]; // G. C. Greubel, Jul 14 2024

Table[2*Binomial[n+3, k+2 +Floor[(n+1)/2]], {n,0,12}, {k,0,Floor[n/2] }]//Flatten (* G. C. Greubel, Jul 14 2024 *)

def A028326(n,k): return 2*binomial(n, k)
flatten([[A028326(n+1,k) for k in range(((n+3)//2), n+1)] for n in range(21)]) # G. C. Greubel, Jul 14 2024

From G. C. Greubel, Jul 14 2024: (Start)
T(n, k) = 2*binomial(n+3, k+2 + floor((n+1)/2)).
Sum_{k=0..floor(n/2)} T(n, k) = A272514(n+3).
Sum_{k=0..n} (-1)^k*T(2*n, k) = 2*A286033(n+2).
Sum_{k=0..n} (-1)^k*T(2*n+1, k) = binomial(2*n+4, n+2) + 2*(-1)^n.
(End)

More terms from James Sellers

A028332 Distinct elements to the right of the central elements of the even-Pascal triangle A028326.

Original entry on oeis.org

2, 6, 8, 20, 10, 30, 12, 70, 42, 14, 112, 56, 16, 252, 168, 72, 18, 420, 240, 90, 924, 660, 330, 110, 22, 1584, 990, 440, 132, 24, 3432, 2574, 1430, 572, 156, 26, 6006, 4004, 2002, 728, 182, 28, 12870, 10010, 2730, 910, 210, 22880, 16016, 8736, 3640

Offset: 0

Mohammad K. Azarian

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A028326, A028327, A028328, A028329, A028330, A028331.

DeleteDuplicates[Table[2*Binomial[n+1, k+1 +Floor[(n+1)/2]], {n,0,30}, {k,0,Floor[n/2]}]//Flatten] (* G. C. Greubel, Jul 14 2024 *)

A028330=flatten([[2*binomial(n+1,k+1+((n+1)//2)) for k in range(1+(n//2))] for n in range(31)])
def a(seq): # order preserving
    nd = [] # no duplicates
    [nd.append(i) for i in seq if not nd.count(i) and i%2==0]
    return nd
a(A028330) # A028332 # G. C. Greubel, Jul 14 2024

More terms from Asher Auel

Duplicated 20 removed by Sean A. Irvine, Dec 29 2019

cp's OEIS Frontend

A028329 Twice central binomial coefficients.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

Extensions

A028327 Elements in the even-Pascal triangle A028326 that are not 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Extensions

A028328 Distinct elements in the even-Pascal triangle A028326.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

SageMath

Extensions

A028331 Elements to the right of the central elements of the even-Pascal triangle A028326 that are not 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A028332 Distinct elements to the right of the central elements of the even-Pascal triangle A028326.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

SageMath

Extensions