A028329 -id:A028329 - cp's OEIS frontend

A379200 G.f. A(x,y) satisfies 1/x = Sum_{n=-oo..+oo} A(x,y)^n * (A(x,y)^n + y)^(n+1), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows.

1, 2, 1, 4, 4, 2, 8, 13, 12, 5, 18, 40, 52, 40, 14, 52, 130, 204, 215, 140, 42, 184, 472, 813, 1004, 896, 504, 132, 688, 1863, 3430, 4588, 4816, 3738, 1848, 429, 2512, 7536, 15016, 21472, 24540, 22656, 15576, 6864, 1430, 8866, 30144, 65880, 102177, 124830, 126801, 104940, 64779, 25740, 4862, 30824, 118420, 284305, 483300, 636750, 693528, 638825, 479908, 268840, 97240, 16796

Offset: 1

Paul D. Hanna, Dec 20 2024

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

			G.f.: A(x,y) = x*(1) + x^2*(2 + y) + x^3*(4 + 4*y + 2*y^2) + x^4*(8 + 13*y + 12*y^2 + 5*y^3) + x^5*(18 + 40*y + 52*y^2 + 40*y^3 + 14*y^4) + x^6*(52 + 130*y + 204*y^2 + 215*y^3 + 140*y^4 + 42*y^5) + x^7*(184 + 472*y + 813*y^2 + 1004*y^3 + 896*y^4 + 504*y^5 + 132*y^6) + x^8*(688 + 1863*y + 3430*y^2 + 4588*y^3 + 4816*y^4 + 3738*y^5 + 1848*y^6 + 429*y^7) + x^9*(2512 + 7536*y + 15016*y^2 + 21472*y^3 + 24540*y^4 + 22656*y^5 + 15576*y^6 + 6864*y^7 + 1430*y^8) + x^10*(8866 + 30144*y + 65880*y^2 + 102177*y^3 + 124830*y^4 + 126801*y^5 + 104940*y^6 + 64779*y^7 + 25740*y^8 + 4862*y^9) + ...
where 1/x = Sum_{n=-oo..+oo} A(x,y)^n * (A(x,y)^n + y)^(n+1).
TRIANGLE.
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for n >= 1, k=0..n-1, begins
n = 1: [1];
n = 2: [2, 1];
n = 3: [4, 4, 2];
n = 4: [8, 13, 12, 5];
n = 5: [18, 40, 52, 40, 14];
n = 6: [52, 130, 204, 215, 140, 42];
n = 7: [184, 472, 813, 1004, 896, 504, 132];
n = 8: [688, 1863, 3430, 4588, 4816, 3738, 1848, 429];
n = 9: [2512, 7536, 15016, 21472, 24540, 22656, 15576, 6864, 1430];
n =10: [8866, 30144, 65880, 102177, 124830, 126801, 104940, 64779, 25740, 4862];
n =11: [30824, 118420, 284305, 483300, 636750, 693528, 638825, 479908, 268840, 97240, 16796];
n =12: [108088, 460746, 1205402, 2242581, 3213584, 3758727, 3731794, 3154866, 2171312, 1113398, 369512, 58786];
  ...
RELATED SEQUENCES.
A000108(n) = T(n+1,n) for n >= 0 (Catalan numbers).
A028329(n) = T(n+2,n) for n >= 0.
A166952(n) = T(n+1,0) for n >= 0 (g.f. F(x) = theta_3(x*F(x))).
A379201(n) = T(n,1) for n >= 2 (column 1).
A379206(n) = T(2*n-1,n-1) for n >= 1 (central terms).
A378264(n) = Sum_{k=0..n-1} T(n,k) for n >= 1.
A379199(n) = Sum_{k=0..n-1} T(n,k) * (-1)^k for n >= 1.
A379202(n) = Sum_{k=0..n-1} T(n,k) * 2^k for n >= 1.
A379203(n) = Sum_{k=0..n-1} T(n,k) * 3^k for n >= 1.
A379204(n) = Sum_{k=0..n-1} T(n,k) * 4^k for n >= 1.
A379205(n) = Sum_{k=0..n-1} T(n,k) * 5^k for n >= 1.
ALTERNATIVE FORMAT.
This triangle may also be presented as a rectangular table like so:
[  1,    1,     2,      5,     14,      42,      132, ...];
[  2,    4,    12,     40,    140,     504,     1848, ...];
[  4,   13,    52,    215,    896,    3738,    15576, ...];
[  8,   40,   204,   1004,   4816,   22656,   104940, ...];
[ 18,  130,   813,   4588,  24540,  126801,   638825, ...];
[ 52,  472,  3430,  21472, 124830,  693528,  3731794, ...];
[184, 1863, 15016, 102177, 636750, 3758727, 21365548, ...];
...

Paul D. Hanna, Table of n, a(n) for n = 1..2628; the initial 72 rows of this triangle.

Cf. A166952 (column 0, y=0), A378264 (row sums), A379201 (column 1), A379206 (central terms).
Cf. A379199 (y=-1), A379202 (y=2), A379203 (y=3), A379204 (y=4), A379205 (y=5).
Cf. A000108 (main diagonal), A028329 (diagonal).

{T(n,k) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + y)^(m+1) ), #V-3); ); polcoef(polcoef(A, n, x), k, y)}
for(n=1,12, for(k=0,n-1, print1(T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x,y)^n * (A(x,y)^n + y)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x,y)^(2*n) * (A(x,y)^n - y)^n.
(3) A(x,y) = x * Sum_{n=-oo..+oo} A(x,y)^(n^2) / (1 + y*A(x,y)^(n+1))^n.
(4) A(x,y) = x * Sum_{n=-oo..+oo} A(x,y)^(n^2) / (1 - y*A(x,y)^(n+1))^(n+1).
(5) A(B(x,y), y) = x where B(x,y) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + y)^(n+1) ).

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

A028330 Elements to the right of the central elements of the even-Pascal triangle A028326.

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian

			This sequence represents the following portion of A028330(n,k), with x being the elements of A028329(n):
  x;
  .,  2;
  .,  x,  2;
  .,  .,  6,  2;
  .,  .,  x,  8,  2;
  .,  .,  ., 20, 10,   2;
  .,  .,  .,  x, 30,  12,   2;
  .,  .,  .,  ., 70,  42,  14,    2;
  .,  .,  .,  .,  x, 112,  56,   16,   2;
  .,  .,  .,  .,  ., 252, 168,   72,  18,   2;
  .,  .,  .,  .,  .,   x, 420,  240,  90,  20,   2;
  .,  .,  .,  .,  .,   ., 924,  660, 330, 110,  22,  2;
  .,  .,  .,  .,  .,   .,   x, 1584, 990, 440, 132, 24, 2;
As an irregular triangle:
    2;
    2;
    6,   2;
    8,   2;
   20,  10,   2;
   30,  12,   2;
   70,  42,  14,   2;
  112,  56,  16,   2;
  252, 168,  72,  18,  2;
  420, 240,  90,  20,  2;
  924, 660, 330, 110, 22,  2;

G. C. Greubel, Table of n, a(n) for n = 0..2549

Cf. A028326, A028327, A028328, A028329, A028331, A028332.

Cf. A001405, A014413, A058622, A063886, A202736.

[[2*Binomial(n,k): k in [Floor((n+2)/2)..n]]: n in [1..12]]; // G. C. Greubel, Jul 14 2024

Table[2*Binomial[n+1, k+1 +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,k) for k in range(((n+2)//2), n+1)] for n in range(1,21)]) # G. C. Greubel, Jul 14 2024

a(n) = 2 * A014413(n). - Sean A. Irvine, Dec 29 2019
From G. C. Greubel, Jul 14 2024: (Start)
T(n, k) = 2*binomial(n+1, k+1 + floor((n+1)/2)) for n >= 0, 0 <= k <= floor(n/2).
Sum_{k=0..floor(n/2)} T(n, k) = A202736(n+1) = 2*A058622(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n, k) = 2*A001405(n) = A063886(n+1). (End)

More terms from James Sellers

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

A067804 Triangle read by rows: T(n,k) is the number of walks (each step +-1) of length 2n which have a cumulative value of 0 last at step 2k.

Original entry on oeis.org

1, 2, 2, 6, 4, 6, 20, 12, 12, 20, 70, 40, 36, 40, 70, 252, 140, 120, 120, 140, 252, 924, 504, 420, 400, 420, 504, 924, 3432, 1848, 1512, 1400, 1400, 1512, 1848, 3432, 12870, 6864, 5544, 5040, 4900, 5040, 5544, 6864, 12870, 48620, 25740, 20592, 18480

Offset: 0

Henry Bottomley, Feb 07 2002

Since the triangle is symmetric, the probability that a one-dimensional random walk returns to the origin at all in the steps m through to 2m is 1/2 (for m odd).

Diagonal sums are A106183. - Paul Barry, Apr 24 2005

			Triangle begins:
    1;
    2,   2;
    6,   4,   6;
   20,  12,  12,  20;
   70,  40,  36,  40,  70;
  252, 140, 120, 120, 140, 252;
  ...
For a walk of length 4 (=2*2), 6 are only ever 0 at step 0, 4 are zero at step 2 but not step 4 and 6 are 0 at step 4.
For n=3,k=2, T(3,2)=12 since there are 12 monotonic paths from (0,0) to (2,2) and then on to (3,3). Using E for eastward steps and N for northward steps, the 12 paths are given by EENNNE, ENENNE, ENNENE, NNEENE, NENENE, NEENNE, EENNEN, ENENEN, ENNEEN, NNEEEN, NENEEN, NEENEN. - _Dennis P. Walsh_, Mar 23 2012

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 79, ex. 3f.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
B. C. Carlson, Power series for inverse Jacobian elliptic functions, Math. Comp., 77 (2008), 1615-1621, see p. 1617, equation (2.20).
C. M. Grinstead and J. L. Snell, Introduction to Probability p. 482.
R. P. Kelisky, Inverse elliptic functions and Legendre polynomials, Amer. Math. Monthly 66 (1959), pp. 480-483. MR0103993 (21 #2755).
Michael Z. Spivey, A Combinatorial Proof for the Alternating Convolution of the Central Binomial Coefficients, The American Mathematical Monthly 121.6 (2014): 537-540. [Suggested by _Roger L. Bagula_, Jun 21 2014]

Columns include A000984, A028329. Central diagonal is A002894.

Cf. A002426, A120406.

/* As triangle */ [[Binomial(2*k, k)*Binomial(2*n-2*k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 19 2015

Table[Table[Binomial[2k,k]Binomial[2(n-k),n-k],{k,0,n}],{n,0,10}]//Grid  (* Geoffrey Critzer, Jun 30 2013 *)
T[ n_, k_] := SeriesCoefficient[ D[ InverseJacobiSN[2 x, m] / 2, x], {x, 0, 2 n}, {m, 0, k}]; (* Michael Somos, May 06 2017 *)

T(n, k) = binomial(2*k, k) * binomial(2*n-2*k, n-k) /* Michael Somos, Aug 20 2005 */

T(n, k) = C(2k, k)*C(2n-2k, n-k) = C(2n, n)*C(n, k)^2/C(2n, 2k) = A000984(k)*A000984(n-k) = A000984(n)*A008459(n, k)/A007318(2n, 2k).
Row sums are 4^n = A000302(n).
G.f.: A(x,y) = 1/sqrt((1-4*x)*(1-4*x*y)). - Vladeta Jovovic, Dec 12 2003
Sum{k>=0} T(n, k)*(-3)^k = (-4)^n * A002426(n). Sum_{k>=0} T(n, k)/(2*k+1) = 2^(4*n)/((2*n+1)*C(2*n, n)). - Philippe Deléham, Dec 31 2003
O.g.f.: A(x,y) = 1 + x*d/dx(log(B(x,y))), where B(x,y) is the o.g.f. of A120406. - Peter Bala, Jul 17 2015

A084868 Main diagonal of symmetric square table A084867, in which the antidiagonal sums (A006012) form the first row shifted left.

Original entry on oeis.org

1, 2, 8, 36, 168, 796, 3800, 18216, 87536, 421292, 2029592, 9784088, 47187536, 227651352, 1098523504, 5301727824, 25590307552, 123529362124, 596337248024, 2878947861432, 13899229883024, 67105641925064, 323993230750672

Offset: 0

Paul D. Hanna, Jun 10 2003, Jun 11 2003

The Hankel transform (see A001906 for definition) of this sequence is A000302 (powers of 4): 1, 4, 16, 64, 256, 1024, ... - Philippe Deléham, Aug 17 2005

			1 + 2*x + 8*x^2 + 36*x^3 + 168*x^4 + 796*x^5 + 3800*x^6 + 18216*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vincent Pilaud, V Pons, Permutrees, arXiv preprint arXiv:1606.09643, 2016

Cf. A006012, A011782, A028329, A084867, A026671, A081696.

1/(1-x/(sqrt(1/4-x))): series(%,x,23): seq(coeff(%,x,n),n=0..22); # Peter Luschny, Feb 06 2017

Table[SeriesCoefficient[((1-4*x)+2*x*Sqrt[1-4*x])/(1-4*x-4*x^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

{a(n) = if( n<0, 0, polcoeff((1 - 4*x + 2*x * sqrt(1 - 4*x + x * O(x^n))) /(1 - 4*x - 4*x^2), n))} /* Michael Somos, Jan 05 2012 */

Differential equation: (16*x^3 + 12*x^2 - 8*x + 1) * x*(d/dx)A(x) + (8x^3 - 12*x^2 + 6*x - 1) * A(x) + (8x^2 - 6*x + 1) = 0.
G.f.: ((1 - 4*x) + 2*x * sqrt(1 - 4*x)) / (1 - 4*x - 4*x^2). a(n) * (n-1) = a(n-1) * (8*n - 14) - a(n-2) * 12*(n-3) - a(n-3) * 8*(2*n - 5), n > 2. Hankel number wall zig-zag diagonal is A011782. - Michael Somos, Sep 14 2003
INVERT transform of A028329 (offset 1). - Michael Somos, Jan 05 2012
G.f.: (1-2*x*f(x))/(1-2*x*f(x)-2*x) where f(x) is the g.f. of A000108 (Catalan numbers). - Philippe Deléham, Jan 30 2012
a(n) ~ (1-1/sqrt(2))*(2+2*sqrt(2))^n. - Vaclav Kotesovec, Oct 14 2012
From Peter Bala, Feb 05 2017: (Start)
G.f: sqrt(1 - 4*x)/(sqrt(1 - 4*x) - 2*x) =  1/(1 - 2*x/(1 - 2*x/(1 - x/(1 - x/(1 - x/(1 - ...)))))) (continued fraction).  Cf. A026671, A081696.
Catalan transform of A006012, that is, equals A106566*A006012, as noted by R. J. Mathar. (End)

A247817 Sum(4^k, k=2..n).

Original entry on oeis.org

0, 16, 80, 336, 1360, 5456, 21840, 87376, 349520, 1398096, 5592400, 22369616, 89478480, 357913936, 1431655760, 5726623056, 22906492240, 91625968976, 366503875920, 1466015503696, 5864062014800, 23456248059216, 93824992236880, 375299968947536, 1501199875790160

Offset: 1

Vincenzo Librandi, Sep 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. Sum(h^k,k=2..n): A028329 (h=2), A168569 (h=3), this sequence (h=4), A168571 (h=5), A247840 (h=6), A168572 (h=7), A247841 (h=8), A247842 (h=9), A124166 (h=10).

[0] cat [&+[4^k: k in [2..n]]: n in [2..30]];

Magma
```
[(4^(n+1)-16)/3: n in [1..30]];
```

RecurrenceTable[{a[1] == 0, a[n] == a[n-1] + 4^n}, a, {n, 30}] (* or *) CoefficientList[ Series[16 x / ((1 - x) (1 - 4 x)),{x, 0, 40}], x]
LinearRecurrence[{5,-4},{0,16},30] (* Harvey P. Dale, Feb 19 2023 *)

a(n) = sum(k=2, n, 4^k); \\ Michel Marcus, Sep 25 2014

G.f.: 16*x^2/((1-x)*(1-4*x)).
a(n) = a(n-1) + 4^n = (4^(n+1) - 16)/3 = 5*a(n-1) - 4*a(n-2).
a(n) = A080674(n) - 4. - Michel Marcus, Sep 25 2014

A240530 a(n) = 4(2n)! / (n!)^2.

Original entry on oeis.org

4, 8, 24, 80, 280, 1008, 3696, 13728, 51480, 194480, 739024, 2821728, 10816624, 41602400, 160466400, 620470080, 2404321560, 9334424880, 36300541200, 141381055200, 551386115280, 2153031497760, 8416395854880, 32933722910400, 128990414732400

Offset: 0

Vincenzo Librandi, Apr 12 2014

nonn

Apart from first term, the same as A146534. - Arkadiusz Wesolowski, Apr 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Pedrazzi and G. Goldoni, Un labirinto cartesiano (A Cartesian Labyrinth), Archimede, Anno XXXVIII, Jan-Mar 1986, p. 41 (in Italian).

Cf. A000984, A028329, A146534.

List([0..30], n-> 4*Binomial(2*n,n) ); # G. C. Greubel, Dec 19 2019

Magma
```
[4*Binomial (2*n,n): n in [0..30]];
```

seq( 4*binomial(2*n,n), n=0..30); # G. C. Greubel, Dec 19 2019

Table[4*(2*n)!/(n!)^2, {n, 0, 40}] (* or *) CoefficientList[Series[4/Sqrt[1 - 4 x], {x, 0, 50}], x]

vector(31, n, 4*binomial(2*n-2, n-1)) \\ G. C. Greubel, Dec 19 2019

[4*binomial(2*n,n) for n in (0..30)] # G. C. Greubel, Dec 19 2019

G.f.: 4/sqrt(1-4*x).
a(n) = 4*binomial(2*n, n) = 4*A000984(n) = 2*A028329(n).
D-finite with recurrence: n*a(n) - 2*(2*n-1)*a(n-1) = 0 for n > 0.

cp's OEIS Frontend

A379200 G.f. A(x,y) satisfies 1/x = Sum_{n=-oo..+oo} A(x,y)^n * (A(x,y)^n + y)^(n+1), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows.

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

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A028328 Distinct elements in the even-Pascal triangle A028326.

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A028330 Elements to the right of the central elements of the even-Pascal triangle A028326.

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

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

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A067804 Triangle read by rows: T(n,k) is the number of walks (each step +-1) of length 2n which have a cumulative value of 0 last at step 2k.

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A240530 a(n) = 4(2n)! / (n!)^2.