A045741 -id:A045741 - cp's OEIS frontend

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

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)

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)

A127537 Triangle read by rows: T(n,k) (n >= 2, 1 <= k <= 2n-3) is the number of non-crossing connected graphs on n nodes on a circle, having k edges. Rows are indexed 2,3,4,...; columns are indexed 0,1,2,....

Original entry on oeis.org

1, 0, 3, 1, 0, 0, 12, 9, 2, 0, 0, 0, 55, 66, 30, 5, 0, 0, 0, 0, 273, 455, 315, 105, 14, 0, 0, 0, 0, 0, 1428, 3060, 2856, 1428, 378, 42, 0, 0, 0, 0, 0, 0, 7752, 20349, 23940, 15960, 6300, 1386, 132, 0, 0, 0, 0, 0, 0, 0, 43263, 134596, 191268, 159390, 83490, 27324, 5148, 429

Offset: 2

Emeric Deutsch, Jan 24 2007

Row n contains 2n-3 terms, the first n-2 of which are equal to 0.
T(n,n-1) = A001764(n-1). T(n,2n-3) = A000108(n-2) (the Catalan numbers).
T(n,k) = A089434(n,k+1-n).
Sum_{k=n-1..2n-3} k*T(n,k) = A045741(n).
Sum_{n=k..2k-2} T(n,k) = A065065(k).

			Triangle starts:
  1;
  0,  3,  1;
  0,  0, 12,  9,  2;
  0,  0,  0, 55, 66, 30,  5;

C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
C. Domb & A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)
C. Domb & A. J. Barrett, Notes on Table 2 in "Enumeration of ladder graphs", Discrete Math. 9 (1974), 55. (Annotated scanned copy)
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

Cf. A000108, A001764, A045741, A065065, A089434.

T:=(n,k)->binomial(3*n-3,n+k)*binomial(k-1,k-n+1)/(n-1): for n from 2 to 10 do seq(T(n,k),k=1..2*n-3) od; # yields sequence in triangular form

T[n_, k_] := Binomial[3n - 3, n + k] Binomial[k - 1, k - n + 1]/(n - 1);
Table[T[n, k], {n, 2, 10}, {k, 1, 2n - 3}] // Flatten (* Jean-François Alcover, Jul 29 2018 *)

T(n,k) = C(3n-3,n+k)C(k-1,k-n+1)/(n-1) (n >= 2, 0 <= k <= 2n-3).

G.f.: G=G(t,z) satisfies tG^3 + tG^2 - z(1+2t)G + z^2*(1+t) = 0.

Keyword tabl changed to tabf by Michel Marcus, Apr 09 2013

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)

cp's OEIS Frontend

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

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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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A127537 Triangle read by rows: T(n,k) (n >= 2, 1 <= k <= 2n-3) is the number of non-crossing connected graphs on n nodes on a circle, having k edges. Rows are indexed 2,3,4,...; columns are indexed 0,1,2,....

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

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

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