A244038 -id:A244038 - cp's OEIS frontend

A160906 Row sums of A159841.

1, 5, 29, 176, 1093, 6885, 43796, 280600, 1807781, 11698223, 75973189, 494889092, 3231947596, 21153123932, 138712176296, 911137377456, 5993760282021, 39481335979779, 260377117268087, 1719026098532296, 11360252318843933, 75141910203168229, 497431016774189912

Offset: 0

R. J. Mathar, May 29 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A075273, A159841.
Cf. A005809, A006256, A183160, A226751.
Cf. A000302, A386811, A386812.
Cf. A244038, A383326.

A160906 := proc(n) add( A159841(n,k), k=0..n) ; end:
seq(A160906(n), n=0..20) ;

Table[Sum[Binomial[3*n+1, 2*n+k+1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 25 2017 *)

a(n) = sum(k=0, n, binomial(3*n+1, 2*n+k+1)); \\ Michel Marcus, Oct 31 2017

a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],-1)
[simplify(a(n)) for n in range(21)] # Peter Luschny, May 19 2015

a(n) = Sum_{k=0..n} A159841(n,k).
Conjecture: a(2n+1) = A075273(3n).
a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],-1). - Peter Luschny, May 19 2015
Conjecture: 2*n*(2*n-1)*(5*n-4)*a(n) +(-295*n^3+451*n^2-130*n-24)*a(n-1) +24*(5*n+1)*(3*n-4)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Jul 20 2016
a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(2*n+1)). - Ilya Gutkovskiy, Oct 25 2017
a(n) ~ 3^(3*n + 3/2) / (sqrt(Pi*n) * 2^(2*n + 1)). - Vaclav Kotesovec, Oct 25 2017
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+k,k). - Seiichi Manyama, Aug 03 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k). - Seiichi Manyama, Aug 07 2025
G.f.: g^2/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 12 2025
G.f.: B(x)^2/(1 + (B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g*(6-g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 16 2025

A045741 Number of edges in all noncrossing connected graphs on n nodes on a circle.

Original entry on oeis.org

1, 9, 82, 765, 7266, 69930, 679764, 6659037, 65635570, 650194974, 6467730204, 64562259762, 646399361076, 6488447895540, 65276186864232, 657998685456093, 6644370824416530, 67198463606576790, 680568874690989900

Offset: 2

Emeric Deutsch

nonn

			a(3)=9; indeed, with vertices u, v, w, the noncrossing connected graphs are {uv,uw}, {vu, vw}, {wu, wv}, and {uv, vw, wu} with a total of 9 edges.

G. C. Greubel, Table of n, a(n) for n = 2..980
Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, On the congruences of some combinatorial numbers, Stud. Appl. Math. vol. 116 (2006) pp. 135-144
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656, 2014

Cf. A007297, A244038, A244039.

A045741 := proc(n) local k ; add(binomial(3*n-3,n+k)*binomial(k,n-1),k=0..2*n-3) ; end: seq(A045741(n),n=2..20) ; # R. J. Mathar, Feb 27 2008

Table[Sum[k*Binomial[3*n - 3, n + k]*Binomial[k - 1, k - n + 1], {k, n - 1, 2*n}]/(n - 1), {n,2,50}] (* G. C. Greubel, Jan 30 2017 *)

for(n=2,50, print1(sum(k=n-1,2*n, k*binomial(3*n-3,n+k)* binomial(k-1,k-n+1))/(n-1), ", ")) \\ G. C. Greubel, Jan 30 2017

a(n) = Sum_{k = n-1 .. 2*n} (k*binomial(3*n-3, n+k)*binomial(k-1, k-n+1))/(n-1).
a(n) = 1 mod 3 if n in A103457; a(n) = 0 mod 3 otherwise [Eu et al.]. - R. J. Mathar, Feb 27 2008
Recurrence: (n-2)*(n-1)*(6*n-17)*a(n) = 18*(n-2)*a(n-1) + 12*(3*n-8)*(3*n-7)*(6*n-11)*a(n-2). - Vaclav Kotesovec, Dec 29 2012
a(n) ~ (sqrt(3)-1)/sqrt(Pi) * (2^(n-5/2)*3^(3*n/2-3/2))/sqrt(n). - Vaclav Kotesovec, Dec 29 2012
A244038(n) = a(n) + A244039(n) [Gessel]. - N. J. A. Sloane, Jun 28 2014

A244039 a(n) = 2^(2n-1) ( binomial(3n/2,n) + binomial((3n-1)/2,n) ).

Original entry on oeis.org

1, 5, 39, 338, 3075, 28770, 274134, 2645844, 25781283, 253068530, 2498678754, 24788450076, 246889978062, 2467197059124, 24725226928140, 248396412496488, 2500825206700323, 25225687837101330, 254877697946626410, 2579123090162503500, 26133512970919973850, 265126176290618366460

Offset: 0

N. J. A. Sloane, Jun 28 2014

nonn

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

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_3(n).

Cf. A045741, A091527, A244038, A244469.

[Round(2^(2*n-1)*( Gamma(3*n/2+1)/Gamma(n/2+1) + Gamma((3*n+1)/2)/Gamma((n+1)/2) )/Factorial(n)): n in [0..25]]; // G. C. Greubel, Aug 20 2019

a := n -> 2^(2*n-1)*(binomial(3*n/2,n) + binomial((3*n-1)/2,n));
seq(a(n), n=0..25);

Table[2^(2n-1)*(Binomial[3n/2, n] + Binomial[(3n-1)/2, n]), {n, 0, 25}] (* Vincenzo Librandi, Jun 29 2014 *)

a(n) = 2^(2*n-1)*(binomial(3*n/2, n) + binomial((3*n-1)/2, n));
vector(25, n, n--; a(n)) \\ G. C. Greubel, Aug 20 2019

[2^(2*n-1)*(binomial(3*n/2, n) + binomial((3*n-1)/2, n)) for n in (0..25)] # G. C. Greubel, Aug 20 2019

From Peter Bala, Mar 04 2022: (Start)
a(n) = [x^n] ( (1 + 2*x)^3/(1 + x) )^n. Cf. A091527.
a(n) = Sum_{k = 0..n} (-1)^k * 2^(n-k) * binomial(3*n,n-k) * binomial(n+k-1,k).
n*(n-1)*(6*n-11)*a(n) = - 18*(n-1)*a(n-1) + 12*(3*n-4)*(3*n-5)*(6*n-5)*a(n-2) with a(0) = 1 and a(1) = 5.
The o.g.f. A(x) = 1 + 5*x + 39*x^2 + 338*x^3 + ... is the diagonal of the bivariate rational function 1/(1 - t*(1 + 2*x)^3/(1 + x)) and hence is an algebraic function over the field of rational functions Q(x) by Stanley 1999, Theorem 6.33, p. 197.
Calculation gives (1 - 108*x^2)*A(x)^3 - (1 + 9*x)*A(x) = x.
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. (End)
a(n) = 2^n*binomial(3*n, n)*hypergeom([-n, n], [2*n + 1], 1/2). - Peter Luschny, Mar 07 2022
From Seiichi Manyama, Aug 08 2025: (Start)
a(n) = Sum_{k=0..n} binomial(3*n,k) * binomial(2*n-k,n-k).
a(n) = [x^n] (1+x)^(3*n)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^n * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(2*n-k-1,n-k). (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

A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)binomial(2n - k, n).

Original entry on oeis.org

1, 2, 2, 6, 9, 6, 20, 40, 40, 20, 70, 175, 225, 175, 70, 252, 756, 1176, 1176, 756, 252, 924, 3234, 5880, 7056, 5880, 3234, 924, 3432, 13728, 28512, 39600, 39600, 28512, 13728, 3432, 12870, 57915, 135135, 212355, 245025, 212355, 135135, 57915, 12870

Offset: 0

Peter Luschny, Mar 21 2024

The main diagonal and column 0 of the triangle are the central binomial coefficients, which are the sums of the squares of Pascal's triangle entries. This sum representation can be generalized, and all terms can be seen as sums of coefficients of some polynomials. (See the Example section.)

To see this, consider T(n, k) as the value of the polynomials P(n, k)(x) at x = 1, where P(n, k)(x) = H([-n, -k], [1], x)*H([-n, -n + k], [1], x) and H denotes the hypergeometric sum 2F1. For instance column 0 is given by the row sums of A008459, and column 1 by the row sums of A371401.

			Triangle starts:
[0]    1;
[1]    2,     2;
[2]    6,     9,     6;
[3]   20,    40,    40,    20;
[4]   70,   175,   225,   175,    70;
[5]  252,   756,  1176,  1176,   756,   252;
[6]  924,  3234,  5880,  7056,  5880,  3234,   924;
[7] 3432, 13728, 28512, 39600, 39600, 28512, 13728, 3432;
.
Because of the symmetry, only the sum representation of terms with k <= n/2 are shown.
0:                 [1]
1:               [1+1]
2:             [1+4+1],               [1+4+4]
3:           [1+9+9+1],            [1+9+21+9]
4:      [1+16+36+16+1],       [1+16+66+76+16],        [1+16+76+96+36]
5: [1+25+100+100+25+1], [1+25+160+340+205+25], [1+25+190+460+400+100]

Column 0 and main diagonal are A000984.
Column 1 and subdiagonal are A097070.
Row sums are A045721.
The even bisection of the alternating row sums is A005809.
The central terms are A188662.
Cf. A046899, A092392, A371395, A244038, A371399.
Cf. A008459, A371401.

T := (n, k) -> binomial(k + n, k) * binomial(2*n - k, n):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);

T[n_, k_] := Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, -n +k, 1, 1];
Table[T[n, k], {n, 0, 7}, {k, 0, n}]

T(n, k) = A046899(n, k) * A092392(n, k).
T(n, k) = A046899(n, k) * A046899(n, n - k).
T(n, k) = A092392(n, k) * A092392(n, n - k).
T(n, k) = A371395(n, k) * (n + 1).
T(n, k) = hypergeom([-n, -k], [1], 1) * hypergeom([-n, -n + k], [1], 1).
2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A244038(n).
2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A371399(n).

A206306 Riordan array (1, x/(1-3x+2x^2)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 7, 6, 1, 0, 15, 23, 9, 1, 0, 31, 72, 48, 12, 1, 0, 63, 201, 198, 82, 15, 1, 0, 127, 522, 699, 420, 125, 18, 1, 0, 255, 1291, 2223, 1795, 765, 177, 21, 1, 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1

Offset: 0

Philippe Deléham, Feb 06 2012

The convolution triangle of the Mersenne numbers A000225. - Peter Luschny, Oct 09 2022

			Triangle begins:
  1;
  0,    1;
  0,    3,    1;
  0,    7,    6,     1;
  0,   15,   23,     9,     1;
  0,   31,   72,    48,    12,     1;
  0,   63,  201,   198,    82,    15,    1;
  0,  127,  522,   699,   420,   125,   18,    1;
  0,  255, 1291,  2223,  1795,   765,  177,   21,   1;
  0,  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;
  0, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Columns: A000007, A000225 (Mersenne numbers), A045618, A055582.
Cf. A008585, A009545, A084938, A062725, A110441, A204089, A204091.
Cf. A045741, A078020, A244038.

function T(n,k) // T = A206306
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 0 then return 0;
  else return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 20 2022

# Uses function PMatrix from A357368.
PMatrix(10, n -> 2^n - 1); # Peter Luschny, Oct 09 2022

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==0, 0, 3*T[n- 1, k] +T[n-1, k-1] -2*T[n-2, k]]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 20 2022 *)

def T(n,k): # T = A206306
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==0): return 0
    else: return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 20 2022

Triangle T(n,k), read by rows, given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Diagonals sums are even-indexed Fibonacci numbers.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A204089(n), A204091(n) for x = 0, 1, 2 respectively.
G.f.: (1-3*x+2*x^)/(1-(3+y)*x+2*x^2).
From Philippe Deléham, Nov 17 2013; corrected Feb 13 2020: (Start)
T(n, n) = 1.
T(n+1, n) = 3n = A008585(n).
T(n+2, n) = A062725(n).
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=T(2,0)=0, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. (End)
From G. C. Greubel, Dec 20 2022: (Start)
Sum_{k=0..n} (-1)^k*T(n,k) = [n=1] - A009545(n).
Sum_{k=0..n} (-2)^k*T(n,k) = [n=1] + A078020(n+1).
T(2*n, n+1) = A045741(n+2), n >= 0.
T(2*n+1, n+1) = A244038(n). (End)

A299855 G.f. C(x)^(1/2) satisfies C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)S(x)^(1/2) = S'(x)C(x)^(1/2) = 72*x.

Original entry on oeis.org

1, 6, -12, 60, -384, 2772, -21504, 175032, -1474560, 12748164, -112459776, 1008263880, -9160359936, 84151254600, -780341870592, 7294711613040, -68670084612096, 650409360439140, -6193772337561600, 59267126633699880, -569566264641454080, 5494909312181603160, -53198968510406983680, 516695227418183158800, -5033085020810678108160

Offset: 0

Paul D. Hanna, Feb 20 2018

sign

a(n) = -(-1)^n * A244038(n) / (3*n-2) for n>=1.

			G.f.: C(x)^(1/2) = 1 + 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
RELATED SERIES.
C(x) = 1 + 12*x + 12*x^2 - 24*x^3 + 96*x^4 - 504*x^5 + 3072*x^6 - 20592*x^7 + 147456*x^8 - 1108536*x^9 + 8650752*x^10 + ...
S(x) = 36*x^2 - 144*x^3 + 864*x^4 - 6048*x^5 + 46080*x^6 - 370656*x^7 + 3096576*x^8 - 26604864*x^9 + 233570304*x^10 + ...
sqrt(S(x)) = 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
where C(x)^(1/2) - S(x)^(1/2) = 1
and C'*sqrt(S) = S'*sqrt(C) = 72*x.

Cf. A299853, A299854, A244038.

{a(n) = my(C=1, S=x^2); for(i=0, n, C = 1 + intformal( 72*x/sqrt(S +x^3*O(x^n)) ); S = intformal( 72*x/sqrt(C) ) ); polcoeff(sqrt(C), n)}
for(n=0,30,print1(a(n),", "))

{a(n) = if(n==0,1, -(-4)^n * binomial(3*n/2,n) / (3*n-2) )}
for(n=0,30,print1(a(n),", "))

The functions C = C(x) and S = S(x) satisfy:
(1a) sqrt(C) - sqrt(S) = 1.
(1b) C'*sqrt(S) = S'*sqrt(C) = 72*x.
(1c) C' = 72*x/sqrt(S).
(1d) S' = 72*x/sqrt(C).
Integrals.
(2a) C = 1 + Integral 72*x/sqrt(S) dx.
(2b) S = Integral 72*x/sqrt(C) dx.
(2c) C = 1 + Integral S'*sqrt(C/S) dx.
(2d) S = Integral C'*sqrt(S/C) dx.
Exponentials.
(3a) sqrt(C) = exp( Integral 36*x/(C*sqrt(S)) dx ).
(3b) sqrt(S) = 6*x*exp( Integral 36*x/(S*sqrt(C)) - 1/x dx ).
(3c) C - S = exp( Integral 72*x/(C*sqrt(S) + S*sqrt(C)) dx ).
(3d) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).
Functional equations.
(4a) C = 1/3 - 36*x^2 + (2/3)*C^(3/2).
(4b) S = 36*x^2 - (2/3)*S^(3/2).
Explicit solutions.
(5a) C(x) = 1 + Sum_{n>=1} 2*(-4)^n*binomial(3*n/2,n)/((3*n-2)*(3*n-4)) * x^n.
(5b) S(x) = 36*x^2 + Sum_{n>=3} 18*(-4)^n*(3*n-3)*binomial(3*n/2-2,n)/((3*n-4)*(3*n-6)) * x^n.
(5c) sqrt(C(x)) = 1 + Sum_{n>=1} -(-4)^n * binomial(3*n/2,n)/(3*n-2) * x^n.
Formulas for terms.
a(n) = -(-4)^n * binomial(3*n/2,n) / (3*n-2) for n>=1, with a(0) = 1.

A385639 a(n) = Sum_{k=0..n} binomial(4n+1,k) binomial(2*n-k,n-k).

Original entry on oeis.org

1, 7, 69, 748, 8485, 98847, 1171884, 14066808, 170421669, 2079531685, 25520363869, 314653207128, 3894577133356, 48362609654548, 602248101550920, 7517853111444528, 94044248726758821, 1178641094940246897, 14796230460187072719, 186022053254555479500, 2341837809478393341885

Offset: 0

Seiichi Manyama, Aug 07 2025

nonn

Cf. A098430, A383716, A386811.

Cf. A178792, A244038, A386895.

Table[Sum[Binomial[4*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)

a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(2*n-k, n-k));

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(3*n-k,n-k).
a(n) = binomial(2*n, n)*hypergeom([-1-4*n, -n], [-2*n], -1). - Stefano Spezia, Aug 07 2025
a(n) ~ sqrt((187 - 3*sqrt(17)) / (17*Pi*n)) * (51*sqrt(17) - 107)^n / 2^(3*n + 3/2). - Vaclav Kotesovec, Aug 07 2025

A386899 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3n+1,k) binomial(2*n-k,n-k).

Original entry on oeis.org

1, 16, 339, 7840, 189295, 4689216, 118155156, 3013479744, 77557234095, 2010176842960, 52394920516939, 1371957494204544, 36062378503314436, 950984592573500800, 25147592297769065400, 666594977732384307840, 17706778517771676847215, 471217399398861925667760

Offset: 0

Seiichi Manyama, Aug 07 2025

nonn

Cf. A386830, A386900.

Cf. A244038.

a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(3*n+1, k)*binomial(2*n-k, n-k));

a(n) = [x^n] (1+3*x)^(3*n+1)/(1-2*x)^(n+1).
a(n) = [x^n] 1/((1-3*x) * (1-5*x))^(n+1).
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(n+k,k) * binomial(2*n-k,n-k).
a(n) = 2^n*binomial(2*n, n)*hypergeom([-1-3*n, -n], [-2*n], -3/2). - Stefano Spezia, Aug 07 2025

A348618 a(n) = (1+(-1)^n)/24^n(C((3n)/2-1,n))+(1-(-1)^n)/2((C((3n-1)/2,n))(C(3n-1,(3n-1)/2)))/(C(n-1,(n-1)/2)).

Original entry on oeis.org

1, 2, 16, 140, 1280, 12012, 114688, 1108536, 10813440, 106234700, 1049624576, 10418726760, 103817412608, 1037865473400, 10404558274560, 104557533120240, 1052941297385472, 10623352887172620, 107358720517734400, 1086563988284497800, 11011614449734778880

Offset: 0

Vladimir Kruchinin, Oct 25 2021

nonn

Cf. A244038.

a:= n-> ceil(4^n*binomial(3*n/2, n)/3):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 25 2021

a[n_] := If[EvenQ[n], 4^n * Binomial[3*n/2 - 1, n], Binomial[(3*n - 1)/2, n] * Binomial[3*n - 1, (3*n - 1)/2] / Binomial[n - 1, (n - 1)/2]]; Array[a, 18, 0] (* Amiram Eldar, Oct 25 2021 *)

a(n):=if evenp(n) then 4^n*binomial(3*n/2-1,n) else ((binomial((3*n-1)/2,n))*
    (binomial(3*n-1,(3*n-1)/2)))/binomial(n-1,(n-1)/2);

G.f.: (288*x^2*cos(arcsin(216*x^2-1)/3))/(sqrt(432*x^2-46656*x^4)*(2*sin(arcsin(216*x^2-1)/3)+1)).

Conjecture: D-finite with recurrence n*(n-1)*a(n) -12*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Mar 06 2022

cp's OEIS Frontend

A160906 Row sums of A159841.

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A045741 Number of edges in all noncrossing connected graphs on n nodes on a circle.

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A244039 a(n) = 2^(2*n-1) * ( binomial(3*n/2,n) + binomial((3*n-1)/2,n) ).

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

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A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)*binomial(2*n - k, n).

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A206306 Riordan array (1, x/(1-3*x+2*x^2)).

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A299855 G.f. C(x)^(1/2) satisfies C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)*S(x)^(1/2) = S'(x)*C(x)^(1/2) = 72*x.

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A244039 a(n) = 2^(2n-1) ( binomial(3n/2,n) + binomial((3n-1)/2,n) ).

A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)binomial(2n - k, n).

A206306 Riordan array (1, x/(1-3x+2x^2)).

A299855 G.f. C(x)^(1/2) satisfies C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)S(x)^(1/2) = S'(x)C(x)^(1/2) = 72*x.

A385639 a(n) = Sum_{k=0..n} binomial(4n+1,k) binomial(2*n-k,n-k).

A386899 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3n+1,k) binomial(2*n-k,n-k).

A348618 a(n) = (1+(-1)^n)/24^n(C((3n)/2-1,n))+(1-(-1)^n)/2((C((3n-1)/2,n))(C(3n-1,(3n-1)/2)))/(C(n-1,(n-1)/2)).