A098430 -id:A098430 - cp's OEIS frontend

A001025 Powers of 16: a(n) = 16^n.

1, 16, 256, 4096, 65536, 1048576, 16777216, 268435456, 4294967296, 68719476736, 1099511627776, 17592186044416, 281474976710656, 4503599627370496, 72057594037927936, 1152921504606846976, 18446744073709551616, 295147905179352825856, 4722366482869645213696, 75557863725914323419136, 1208925819614629174706176

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 16), L(1, 16), P(1, 16), T(1, 16). Essentially same as Pisot sequences E(16, 256), L(16, 256), P(16, 256), T(16, 256). See A008776 for definitions of Pisot sequences.
Convolution-square (auto-convolution) of A098430. - R. J. Mathar, May 22 2009
Subsequence of A161441: A160700(a(n)) = 1. - Reinhard Zumkeller, Jun 10 2009
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 16-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Muniru A Asiru, Table of n, a(n) for n = 0..820 (terms n = 0..100 from T. D. Noe)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 280
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (16).

Partial sums give A131865.

Cf. A000079, A000302, A001018, A005843, A008586.

List([0..20],n->16^n); # Muniru A Asiru, Nov 07 2018

a001025 = (16 ^)
a001025_list = iterate (* 16) 1  -- Reinhard Zumkeller, Nov 07 2012

A001025:=-1/(-1+16*z); # Simon Plouffe in his 1992 dissertation

Table[4^(2*n), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 01 2009 *)

A001025(n):=16^n$
makelist(A001025(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=1<<(4*n) \\ Charles R Greathouse IV, Feb 01 2012

print([16**n for n in range(20)]) # Stefano Spezia, Nov 10 2018

[lucas_number1(n,16,0) for n in range(1, 18)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-16*x).
E.g.f.: exp(16*x).
From Muniru A Asiru, Nov 07 2018: (Start)
a(n) = 16^n.
a(0) = 1, a(n) = 16*a(n-1). (End)
a(n) = 4^A005843(n) = 2^A008586(n) = A000302(n)^2 = A000079(n)*A001018(n). - Muniru A Asiru, Nov 10 2018
a(n) = ( Sum_{k = 0..n} (2*k + 1)*binomial(2*n + 1, n - k) ) * ( Sum_{k = 0..n} (-1)^k/(2*k + 1)*binomial(2*n + 1, n - k) ). - Peter Bala, Feb 12 2019
a(n) = Sum_{k = 0..2*n} A000984(k) * A000984(2*n-k). - Peter Bala, Aug 23 2025

A386811 a(n) = Sum_{k=0..n} binomial(4*n+1,k).

Original entry on oeis.org

1, 6, 46, 378, 3214, 27896, 245506, 2182396, 19548046, 176142312, 1594831736, 14497410186, 132224930146, 1209397179048, 11088872706188, 101890087382168, 937973964234638, 8649109175873288, 79872298511230120, 738583466508887304, 6837944227813170424

Offset: 0

Seiichi Manyama, Aug 03 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A000302, A160906, A386812.
Cf. A005810, A078995, A147855, A226733, A226761, A385605.
Cf. A098430, A383716.
Cf. A066381.

[&+[Binomial(4*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 21 2025

Table[Sum[Binomial[4*n+1,k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)

a(n) = sum(k=0, n, binomial(4*n+1, k));

a(n) = [x^n] 1/((1-2*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+k,k).
D-finite with recurrence +645*n*(3*n-1)*(3*n-2)*a(n) +8*(-56722*n^3+213090*n^2-305978*n+150255)*a(n-1) +128*(62908*n^3-282348*n^2+385070*n-126735)*a(n-2) +12288*(-2486*n^3+8918*n^2+758*n-18935)*a(n-3) -2949120*(2*n-7)*(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Aug 03 2025
a(n) = 2^(4*n+1) - binomial(4*n+1, n)*(hypergeom([1, -1-3*n], [1+n], -1) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - Seiichi Manyama, Aug 07 2025
a(n) ~ 2^(8*n + 3/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Aug 07 2025
G.f.: g^2/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 12 2025
G.f.: B(x)^2/(1 + (B(x)-1)/2), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(8-2*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 16 2025

A216702 a(n) = Product_{k=1..n} (16 - 4/k).

Original entry on oeis.org

1, 12, 168, 2464, 36960, 561792, 8614144, 132903936, 2060011008, 32044615680, 499896004608, 7816555708416, 122459372765184, 1921670157238272, 30197673899458560, 475110069351481344, 7482983592285831168, 117967035454858985472, 1861257670509997326336

Offset: 0

Michel Lagneau, Sep 16 2012

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Harvey P. Dale, Table of n, a(n) for n = 0..800

Cf. A004988, A049382, A004994, A034385, A098430.

seq(product(16-4/k, k=1.. n), n=0..20);
seq((4^n/n!)*product(4*k+3, k=0.. n-1), n=0..20);

Table[Product[16-4/k,{k,n}],{n,0,20}] (* or *) CoefficientList[ Series[ 1/(1-16*x)^(3/4),{x,0,20}],x] (* Harvey P. Dale, Sep 19 2012 *)

G.f.: 1/(1-16*x)^(3/4). - Harvey P. Dale, Sep 19 2012
From Peter Bala, Sep 24 2023: (Start)
a(n) = 16^n * binomial(n - 1/4, n).
P-recursive: a(n) = 4*(4*n - 1)/n * a(n-1) with a(0) = 1. (End)
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-16)^n * binomial(-3/4, n).
a(n) ~ 1/Gamma(3/4) * 16^n/n^(1/4).
E.g.f.: hypergeom([3/4], [1], 16*x).
a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-3/4, k)* binomial(-3/4, 2*n - k).
(16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
Sum_{k = 0..n} a(k)*a(n-k) = (16^n)/(2*n)! * Product_{k = 1..n} (4*k^2 - 1) =  (16^n)/(2*n)! * A079484(n). (End)

A244038 a(n) = 4^n * binomial(3*n/2,n).

Original entry on oeis.org

1, 6, 48, 420, 3840, 36036, 344064, 3325608, 32440320, 318704100, 3148873728, 31256180280, 311452237824, 3113596420200, 31213674823680, 313672599360720, 3158823892156416, 31870058661517860, 322076161553203200, 3259691964853493400, 33034843349204336640, 335189468043077792760

Offset: 0

N. J. A. Sloane, Jun 28 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014. See f_1(n).

Cf. A045741, A244039.
Cf. A160906, A383326.
Cf. A000302, A098430, A386897.

[Round(4^n*Gamma(3*n/2+1)/(Gamma(n+1)*Gamma(n/2+1))): n in [0..40]]; // G. C. Greubel, Aug 06 2018

f1:=n->4^n*binomial(3*n/2,n); [seq(f1(n),n=0..40)];

Table[4^n Binomial[3 n/2, n], {n, 0, 40}] (* Vincenzo Librandi, Jun 29 2014 *)

{a(n) = if( n<1, n==0, 3^n * polcoeff( serreverse( x / (x+1) / 2 * sqrt((x+3) / (x+1) / 3 + x * O(x^n))), n))}; /* Michael Somos, Jan 27 2018 */

a(n) = A045741(n+1) + A244039(n) [Gessel].
a(n) = [x^n] 1/sqrt(1 - 4*x)^(n+2). - Ilya Gutkovskiy, Oct 10 2017
G.f. A(x) satisfies: A(x)^3 * (1 - 108*x^2) = 3*A(x) - 2. - Michael Somos, Jan 27 2018
a(n) = [x^n] ( (1 + 4*x)^(3/2) )^n. It follows that the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. - Peter Bala, Mar 05 2022
G.f.: 2/(1-2*sin(arcsin(216*x^2-1)/3)). - Vladimir Kruchinin, Oct 06 2022
G.f.: ((3^(5/6)*i + 3^(1/3))*(-18*i*z + sqrt(-324*z^2 + 3))^(1/3) - (3^(5/6)*i - 3^(1/3))*(18*i*z + sqrt(-324*z^2 + 3))^(1/3))/(2*sqrt(-324*z^2 + 3)), where i = sqrt(-1) is the imaginary unit. - Karol A. Penson, Oct 24 2024
From Seiichi Manyama, Aug 07 2025: (Start)
a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x))^(n+1).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(2*n-k,n-k). (End)

A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

Original entry on oeis.org

1, 4, 40, 480, 6240, 84864, 1188096, 16972800, 246105600, 3609548800, 53421322240, 796463349760, 11946950246400, 180123249868800, 2727580640870400, 41459225741230080, 632253192553758720

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..500
A. Straub, V. H. Moll, and T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10).

Cf. A007696.

Expansion of (1-b^2*x)^(-1/b): A000984 (b=2), A004987 (b=3), this sequence (b=4), A034688 (b=5), A004993 (b=6), A034835 (b=7), A034977 (b=8), A035024 (b=9), A035308 (b=10).

CoefficientList[Series[1/Surd[1-16x,4],{x,0,20}],x] (* Harvey P. Dale, Aug 06 2018 *)

a(n) = (4^n/n!)*A007696(n), n >= 1, a(0) := 1, A007696(n) = (4*n-3)!^4 := Product_{j = 1..n} 4*j - 3.
G.f.: (1 - 16*x)^(-1/4).
D-finite with recurrence: n*a(n) + 4*(-4*n + 3)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-16)^n*binomial(-1/4, n).
a(n) ~ Gamma(3/4)/(sqrt(2)*Pi) * 16^n/n^(3/4).
E.g.f.: hypergeom([1/4], [1], 16*x).
a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-1/4, k)* binomial(-1/4, 2*n - k).
(16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
Sum_{k = 0..n} a(k)*a(n-k) = (4^n)*binomial(2*n, n) = A098430.
Sum_{k = 0..2*n} a(k)*a(2*n-k) = (16^n)*binomial(4*n, 2*n). (End)

A386897 a(n) = 4^n * binomial(5*n/2,n).

Original entry on oeis.org

1, 10, 160, 2860, 53760, 1040060, 20500480, 409404600, 8255569920, 167718033340, 3427543285760, 70384350760360, 1451115518361600, 30018413447053080, 622759359440486400, 12951795276279787760, 269947721071617638400, 5637113741080428839100

Offset: 0

Seiichi Manyama, Aug 07 2025

Paolo Xausa, Table of n, a(n) for n = 0..700

Cf. A386812, A386895, A386896, A386898.

Cf. A000302, A098430, A244038.

Table[Sum[2^k *(-1)^(n-k)*Binomial[5*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)
A386897[n_] := 4^n*Binomial[5*n/2, n]; Array[A386897, 20, 0] (* Paolo Xausa, Aug 26 2025 *)

PARI
```
a(n) = 4^n*binomial(5*n/2, n);
```

a(n) == 0 (mod 10) for n > 0.
a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(4*n-k,n-k).
a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(3*n+1).
a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k,k) * binomial(2*n-k,n-k).
a(n) = [x^n] 1/(1-4*x)^(3*n/2+1).
a(n) = [x^n] (1+4*x)^(5*n/2).
a(n) ~ 2^(n - 1/2) * 5^((5*n+1)/2) / (sqrt(Pi*n) * 3^((3*n+1)/2)). - Vaclav Kotesovec, Aug 07 2025
D-finite with recurrence 3*n*(n-1)*(3*n-4) *(3*n-2)*a(n) -20*(5*n-4) *(5*n-8)*(5*n-2) *(5*n-6)*a(n-2)=0. - R. J. Mathar, Aug 21 2025
O.g.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1/3, 1/2, 2/3], (12500*x^2)/27) + 10*x*hypergeom([7/10, 9/10, 11/10, 13/10], [5/6, 7/6, 3/2], (12500*x^2)/27). - Karol A. Penson, Aug 26 2025

A249308 Central terms of triangle A249307.

Original entry on oeis.org

1, 2, 8, 16, 96, 192, 1280, 2560, 17920, 35840, 258048, 516096, 3784704, 7569408, 56229888, 112459776, 843448320, 1686896640, 12745441280, 25490882560, 193730707456, 387461414912, 2958796259328, 5917592518656, 45368209309696, 90736418619392, 697972450918400

Offset: 0

Reinhard Zumkeller, Nov 14 2014

nonn

a(n) = A128014(n) * 2^n = A249307(n,n);

A098430(n) = a(2*n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A249307, A128014, A098430.

Haskell
```
a249308 n = a249307 n n
```

A174301 A symmetrical triangle: T(n,k) = binomial(n, k)*if(floor(n/2) greater than or equal to k then 4^k, otherwise 4^(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 12, 12, 1, 1, 16, 96, 16, 1, 1, 20, 160, 160, 20, 1, 1, 24, 240, 1280, 240, 24, 1, 1, 28, 336, 2240, 2240, 336, 28, 1, 1, 32, 448, 3584, 17920, 3584, 448, 32, 1, 1, 36, 576, 5376, 32256, 32256, 5376, 576, 36, 1, 1, 40, 720, 7680, 53760, 258048, 53760, 7680, 720, 40, 1

Offset: 0

Roger L. Bagula, Mar 15 2010

Row sums are: {1, 2, 10, 26, 130, 362, 1810, 5210, 26050, 76490, ...}.

			Triangle begins:
  1;
  1,  1;
  1,  8,   1;
  1, 12,  12,    1;
  1, 16,  96,   16,     1;
  1, 20, 160,  160,    20,      1;
  1, 24, 240, 1280,   240,     24,     1;
  1, 28, 336, 2240,  2240,    336,    28,    1;
  1, 32, 448, 3584, 17920,   3584,   448,   32,   1;
  1, 36, 576, 5376, 32256,  32256,  5376,  576,  36,  1;
  1, 40, 720, 7680, 53760, 258048, 53760, 7680, 720, 40,  1;

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

Cf. A144463, A144470.

T(2n,n) gives A098430.

[[Floor(n/2) ge k select 4^k*Binomial(n,k) else 4^(n-k)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 15 2019

Table[Binomial[n, m]*If[Floor[n/2]>=m , 4^m, 4^(n-m)], {n,0,10}, {m,0,n} ]//Flatten

{T(n,k) = binomial(n,k)*if(floor(n/2)>=k, 4^k, 4^(n-k))}; \\ G. C. Greubel, Apr 15 2019

def T(n,k):
   if floor(n/2)>=k: return 4^k*binomial(n,k)
   else: return 4^(n-k)*binomial(n,k)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 15 2019

T(n, m) = binomial(n, m)*if(floor(n/2) greater than or equal to m then 4^m, otherwise 4^(n-m)).

Edited by G. C. Greubel, Apr 15 2019

A253665 a(n) = 2^n*n!/(floor(n/2)!)^2.

Original entry on oeis.org

1, 2, 8, 48, 96, 960, 1280, 17920, 17920, 322560, 258048, 5677056, 3784704, 98402304, 56229888, 1686896640, 843448320, 28677242880, 12745441280, 484326768640, 193730707456, 8136689713152, 2958796259328, 136104627929088, 45368209309696, 2268410465484800

Offset: 0

Peter Luschny, Feb 01 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A056040, A253666, A098430.

a := n -> 2^n*n!/iquo(n,2)!^2: seq(a(n), n=0..25);

Table[2^n*n!/Floor[n/2]!^2, {n, 0, 25}] (* Michael De Vlieger, Feb 02 2015 *)
CoefficientList[Series[(1 + 2 (1 - 8 x) x)/(1 - 16 x^2)^(3/2), {x, 0, 20}],x] (* Benedict W. J. Irwin, Aug 15 2016 *)

a(n)=2^n*n!/(n\2)!^2 \\ Charles R Greathouse IV, Aug 25 2016

a(n) = 2^n*A056040(n).
a(2*n) = 4^n*C(2*n, n) = A098430(n).
a(n) = sum(k=0..n, C(n,k)*n!/(floor(n/2)!)^2) = sum(k=0..n, A253666(n,k)).
G.f.: (1+2*(1-8*x)*x)/(1-16*x^2)^(3/2). - Benedict W. J. Irwin, Aug 15 2016

A269796 a(n) = 4^n * A000108(n+1).

Original entry on oeis.org

1, 8, 80, 896, 10752, 135168, 1757184, 23429120, 318636032, 4402970624, 61641588736, 872465563648, 12463793766400, 179478630236160, 2602440138424320, 37965009078190080, 556820133146787840, 8205770383215820800, 121445401671594147840, 1804331681977970196480, 26900945076762464747520

Offset: 0

Michel Marcus, Mar 07 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016. Cf. 2.1.10.

Cf. A000108, A003645, A101600, A098430.

[4^n*Catalan(n+1): n in [0..25]]; // Vincenzo Librandi, Apr 25 2016

Table[4^n CatalanNumber[n + 1], {n, 0, 20}] (* Bruno Berselli, Mar 07 2016 *)

cat(n) = binomial(2*n,n)/(n+1);
a(n) = 4^n*cat(n+1);

[4^n*catalan_number(n+1) for n in (0..20)] # Bruno Berselli, Mar 07 2016

G.f.: (1 - sqrt(1 - 16*x) - 8*x)/(32*x^2). - Bruno Berselli, Mar 07 2016
a(n) = -2^(4*n + 3)*binomial(n + 1/2, -3/2), after Peter Luschny in A000108. - Bruno Berselli, Mar 07 2016
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 52/75 + 512*arcsin(1/4)/(75*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 164/289 + 1536*arcsinh(1/4)/(289*sqrt(17)). (End)
From Karol A. Penson, Aug 26 2025: (Start)
O.g.f.: exp(Sum_{n>=1} A098430(n)*x^n/n).
O.g.f.: exp(Sum_{n>=1} 4^n*(2*n)!*x^n/(n*(n!)^2)).
a(n) = 4^n*binomial(2*n + 2, n + 1)/(n + 2).
a(n) = Integral_{x=0..16} (x^n*sqrt(x)*sqrt(1 - x/16)/(8*Pi)) dx. (End)

cp's OEIS Frontend

A001025 Powers of 16: a(n) = 16^n.

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A386811 a(n) = Sum_{k=0..n} binomial(4*n+1,k).

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A216702 a(n) = Product_{k=1..n} (16 - 4/k).

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Maple

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A244038 a(n) = 4^n * binomial(3*n/2,n).

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Magma

Maple

Mathematica

PARI

Formula

A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

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Mathematica

Formula

A386897 a(n) = 4^n * binomial(5*n/2,n).

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Mathematica

PARI

Formula

A249308 Central terms of triangle A249307.

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