A004331 -id:A004331 - cp's OEIS frontend

A262977 a(n) = binomial(4*n-1,n).

1, 3, 21, 165, 1365, 11628, 100947, 888030, 7888725, 70607460, 635745396, 5752004349, 52251400851, 476260169700, 4353548972850, 39895566894540, 366395202809685, 3371363686069236, 31074067324187580, 286845713747883300, 2651487106659130740, 24539426037817994160

Offset: 0

Vladimir Kruchinin, Oct 06 2015

From Gus Wiseman, Sep 28 2022: (Start)
Also the number of integer compositions of 4n with alternating sum 2n, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These compositions are ranked by A348614. The a(12) = 21 compositions are:
  (6,2)  (1,2,5)  (1,1,5,1)  (1,1,1,1,4)
         (2,2,4)  (2,1,4,1)  (1,1,2,1,3)
         (3,2,3)  (3,1,3,1)  (1,1,3,1,2)
         (4,2,2)  (4,1,2,1)  (1,1,4,1,1)
         (5,2,1)  (5,1,1,1)  (2,1,1,1,3)
                             (2,1,2,1,2)
                             (2,1,3,1,1)
                             (3,1,1,1,2)
                             (3,1,2,1,1)
                             (4,1,1,1,1)
The following pertain to this interpretation:
- The case of partitions is A000712, reverse A006330.
- Allowing any alternating sum gives A013777 (compositions of 4n).
- A011782 counts compositions of n.
- A034871 counts compositions of 2n with alternating sum 2k.
- A097805 counts compositions by alternating (or reverse-alternating) sum.
- A103919 counts partitions by sum and alternating sum (reverse: A344612).
- A345197 counts compositions by length and alternating sum.
(End)

V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Cf. A006632, A002293, A004331, A005810, A052203, A224274, A257633.
Cf. A000346, A000984, A001700, A002458, A025047, A058622, A081294, A294175.
Cf. A088218, A163455, A165817.

[Binomial(4*n-1,n): n in [0..20]]; // Vincenzo Librandi, Oct 06 2015

Table[Binomial[4 n - 1, n], {n, 0, 40}] (* Vincenzo Librandi, Oct 06 2015 *)

B(x):=sum(binomial(4*n-1,n-1)*3/(4*n-1)*x^n,n,1,30);
taylor(x*diff(B(x),x,1)/B(x),x,0,20);

a(n) = binomial(4*n-1,n); \\ Michel Marcus, Oct 06 2015

G.f.: A(x)=x*B'(x)/B(x), where B(x) if g.f. of A006632.
a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*binomial(3*n,k)).
a(n) = 3*A224274(n), for n > 0. - Michel Marcus, Oct 12 2015
From Peter Bala, Nov 04 2015: (Start)
The o.g.f. equals f(x)/g(x), where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A005810 (k = 0), A052203 (k = 1), A257633 (k = 2), A224274 (k = 3) and A004331 (k = 4). (End)
a(n) = [x^n] 1/(1 - x)^(3*n). - Ilya Gutkovskiy, Oct 03 2017
a(n) = A071919(3n-1,n+1) = A097805(4n,n+1). - Gus Wiseman, Sep 28 2022
From Peter Bala, Feb 14 2024: (Start)
a(n) = (-1)^n * binomial(-3*n, n).
a(n) = hypergeom([1 - 3*n, -n], [1], 1).
The g.f. A(x) satisfies A(x/(1 + x)^4) = 1/(1 - 3*x). (End)
a(n) = Sum_{k = 0..n} binomial(2*n+k-1, k)*binomial(2*n-k-1, n-k). - Peter Bala, Sep 16 2024
G.f.: 1/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A224274 a(n) = binomial(4*n,n)/4.

Original entry on oeis.org

1, 7, 55, 455, 3876, 33649, 296010, 2629575, 23535820, 211915132, 1917334783, 17417133617, 158753389900, 1451182990950, 13298522298180, 122131734269895, 1123787895356412, 10358022441395860, 95615237915961100, 883829035553043580, 8179808679272664720, 75788358475481302185

Offset: 1

Gary Detlefs, Apr 02 2013

In general, binomial(k*n,n)/k = binomial(k*n-1,n-1).
Sequences in the OEIS related to this identity are:
. C(2n,n) = A000984, C(2n,n)/2 = A001700;
. C(3n,n) = A005809, C(3n,n)/3 = A025174;
. C(4n,n) = A005810, C(4n,n)/4 = a(n);
. C(5n,n) = A001449, C(5n,n)/5 = A163456;
. C(6n,n) = A004355, C(6n,n)/6 is not in the OEIS.
Conjecture: a(n) == 1 (mod n^3) iff n is an odd prime.
It is known that a(p) == 1(mod p^3) for prime p >= 3. See Mestrovic, Section 3. - Peter Bala, Oct 09 2015

			For n=2, binomial(4*n,n) = binomial(8,2) = 8*7/2 = 28, so a(2) = 28/4 = 7. - _Michael B. Porter_, Jul 12 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000
Dmitry Kruchinin and Vladimir Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.
Romeo Meštrović, Lucas’ theorem: its generalizations, extensions and applications (1878-2014), arXiv:1409.3820v1 [math.NT], 2014.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Cf. A001700, A001764, A025174, A163456, A002293, A069271, A004331, A005810, A052203, A257633, A262977, A227726.

[Binomial(4*n,n) div 4: n in [1..25]]; // Vincenzo Librandi, Jun 03 2015

Maple
```
seq(binomial(4*n,n)/4, n=1..17);
```

Table[Binomial[4 n, n]/4, {n, 30}] (* Vincenzo Librandi, Jun 03 2015 *)

a(n) = binomial(4*n,n)/4; /* Joerg Arndt, Apr 02 2013 */

a(n) = binomial(4*n,n)/4 = A005810(n)/4.
a(n) = binomial(4*n-1,n-1).
G.f.: A(x) = B'(x)/B(x), where B(x) = 1 + x*B(x)^4 is g.f. of A002293. - Vladimir Kruchinin, Aug 13 2015
From Peter Bala, Oct 08 2015: (Start)
a(n) = 1/2*[x^n] (C(x)^2)^n, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A163456.
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 22*x^3 + ... is the o.g.f. for A002293.
exp( 2*Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 9*x^2 + 52*x^3 + ... is the o.g.f. for A069271. (End)
From Peter Bala, Nov 04 2015: (Start)
With an offset of 1, the o.g.f. equals f(x)*g(x)^3, where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A052203 (k = 1), A257633 (k = 2) and A004331 (k = 4). (End)
a(n) = 1/5*[x^n] (1 + x)/(1 - x)^(3*n + 1) = 1/5*[x^n]( 1/C(-x) )^(5*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A227726. - Peter Bala, Jul 12 2016
a(n) ~ 2^(8*n-3/2)*3^(-3*n-1/2)*n^(-1/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016
O.g.f.: A(x) = f(x)/(1 - 3*f(x)), where f(x) = series reversion (x/(1 + x)^4) = x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ... is the o.g.f. of A002293 with the initial term omitted. Cf. A025174. - Peter Bala, Feb 03 2022
Right-hand side of the identities (1/3)*Sum_{k = 0..n} (-1)^(n+k)*C(x*n,n-k)*C((x+3)*n+k-1,k) = C(4*n,n)/4 and (1/4)*Sum_{k = 0..n} (-1)^k*C(x*n,n-k)*C((x-4)*n+k-1,k) = C(4*n,n)/4, both valid for n >= 1 and x arbitrary. - Peter Bala, Feb 28 2022
Right-hand side of the identity (1/3)*Sum_{k = 0..2*n} (-1)^k*binomial(5*n-k-1,2*n-k)*binomial(3*n+k-1,k) = binomial(4*n,n)/4. - Peter Bala, Mar 09 2022
a(n) = [x^n] G(x)^n, where G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ... is the g.f. of A001764. - Peter Bala, Oct 17 2024

A004319 a(n) = binomial(3*n, n - 1).

Original entry on oeis.org

1, 6, 36, 220, 1365, 8568, 54264, 346104, 2220075, 14307150, 92561040, 600805296, 3910797436, 25518731280, 166871334960, 1093260079344, 7174519270695, 47153358767970, 310325523515700, 2044802197953900, 13488561475572645, 89067326568860640, 588671286046028640

Offset: 1

N. J. A. Sloane

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Milan Janjic, Two Enumerative Functions.
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Cf. A045721 (k=1), A025174 (k=2), A236194 (k=4), A013698 (k=5), A165817 (k=-1), A117671 (k=-2).

Cf. A000139, A001764, A005809, A006013, A236194, A001791, A004331, A004343, A004356, A004369, A004382.

A004319 := proc(n)
binomial(3*n,n-1);
end proc: # R. J. Mathar, Aug 10 2015

Table[Binomial[3n, n - 1], {n, 20}] (* Harvey P. Dale, Sep 21 2011 *)

a(n):=sum((binomial(3*i-1,2*i-1)*binomial(3*n-3*i-3,2*n-2*i-2))/(2*n-2*i-1),i,1,n-1)/2; /* Vladimir Kruchinin, May 15 2013 */

vector(30, n, binomial(3*n, n-1)) \\ Altug Alkan, Nov 04 2015

G.f.: (g-1)/(1-3*z*g^2), where g = g(z) is given by g = 1 + z*g^3, g(0) = 1, i.e. (in Maple notation), g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
a(n) = Sum_{i=0..n-1} binomial(i+2*n, i). - Ralf Stephan, Jun 03 2005
D-finite with recurrence -2*(2*n+1)*(n-1)*a(n) + 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013
a(n) = (1/2) * Sum_{i=1..n-1} binomial(3*i - 1, 2*i - 1)*binomial(3*n - 3*i - 3, 2*n - 2*i - 2)/(2*n - 2*i - 1). - Vladimir Kruchinin, May 15 2013
G.f.: x*hypergeom2F1(5/3, 4/3; 5/2; 27x/4). - R. J. Mathar, Aug 10 2015
a(n) = n*A001764(n). - R. J. Mathar, Aug 10 2015
From Peter Bala, Nov 04 2015: (Start)
With offset 0, the o.g.f. equals f(x)*g(x)^3, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k, n). See the cross-references. (End)
G.f.: cos(t)/(2*sqrt(1 - (27*x)/4)) - sin(t)/(sqrt(3)*sqrt(x)), where t = arcsin((sqrt(27*x))/2)/3. - Vladimir Kruchinin, May 13 2016
a(n) = [x^(2*n+1)] 1/(1 - x)^n. - Ilya Gutkovskiy, Oct 10 2017
a(n) = binomial(n+1, 2) * A000139(n). - F. Chapoton, Feb 23 2024

A052203 a(n) = (4n+1)*binomial(4n,n)/(3n+1).

Original entry on oeis.org

1, 5, 36, 286, 2380, 20349, 177100, 1560780, 13884156, 124403620, 1121099408, 10150595910, 92263734836, 841392966470, 7694644696200, 70539987842520, 648045936942300, 5964720367660956, 54991682779774384, 507749884105448600, 4694436188839116720

Offset: 0

Barry E. Williams, Jan 28 2000

Central terms of the triangles in A122366 and A111418. - Reinhard Zumkeller, Aug 30 2006 and Mar 14 2014

a(n) is the number of paths from (0,0) to (4n,n), taking north and east steps while avoiding exactly 2 consecutive north steps. - Shanzhen Gao, Apr 15 2010

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

Cf. A002293, A051944, A004331, A005810, A224274, A257633, A262977.

a052203 n = a122366 (2 * n) n  -- Reinhard Zumkeller, Mar 14 2014

[Binomial(4*n+1, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

Table[Binomial[4 n + 1, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)

vector(30, n, n--; (4*n+1)*binomial(4*n,n)/(3*n+1)) \\ Altug Alkan, Nov 05 2015

a(n) = C(4n+1, n); a(n) is asymptotic to c/sqrt(n)*(256/27)^n with c=0.614... - Benoit Cloitre, Jan 27 2003 [c = 2^(5/2)/(3^(3/2)*sqrt(Pi)) = 0.61421182128... - Vaclav Kotesovec, Feb 14 2019]
G.f.: g^2/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
G.f.: hypergeom([1/2, 3/4, 5/4], [2/3, 4/3], (256/27)*x). - Robert Israel, Aug 07 2014
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Nov 26 2012
From Peter Bala, Nov 04 2015: (Start)
The o.g.f. equals f(x)*g(x), where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293.
More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A257633 (k = 2), A224274 (k = 3) and A004331 (k = 4). (End)
a(n) = [x^n] 1/(1 - x)^(3*n+2). - Ilya Gutkovskiy, Oct 03 2017
a(n) = Sum_{k = 0..n} binomial(2*n+k+1, k)*binomial(2*n-k-1, n-k). - Peter Bala, Sep 17 2024

More terms from James Sellers, Jan 31 2000

A348614 Numbers k such that the k-th composition in standard order has sum equal to twice its alternating sum.

Original entry on oeis.org

0, 9, 11, 14, 130, 133, 135, 138, 141, 143, 148, 153, 155, 158, 168, 177, 179, 182, 188, 208, 225, 227, 230, 236, 248, 2052, 2057, 2059, 2062, 2066, 2069, 2071, 2074, 2077, 2079, 2084, 2089, 2091, 2094, 2098, 2101, 2103, 2106, 2109, 2111, 2120, 2129, 2131

Offset: 1

Gus Wiseman, Oct 29 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The terms together with their binary indices begin:
    0: ()
    9: (3,1)
   11: (2,1,1)
   14: (1,1,2)
  130: (6,2)
  133: (5,2,1)
  135: (5,1,1,1)
  138: (4,2,2)
  141: (4,1,2,1)
  143: (4,1,1,1,1)
  148: (3,2,3)
  153: (3,1,3,1)
  155: (3,1,2,1,1)
  158: (3,1,1,1,2)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The unordered case (partitions) is counted by A000712, reverse A006330.
These compositions are counted by A262977.
Except for 0, a subset of A345917 (which is itself a subset of A345913).
A000346 = even-length compositions with alt sum != 0, complement A001700.
A011782 counts compositions.
A025047 counts wiggly compositions, ranked by A345167.
A034871 counts compositions of 2n with alternating sum 2k.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
A345197 counts compositions by length and alternating sum.
Cf. A000984, A002458, A004331, A005810, A224274.
Cf. A008549, A013777, A027306, A058622, A088218, A114121, A120452, A126869, A163493, A294175, A344604.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Total[stc[#]]==2*ats[stc[#]]&]

A023827 Sum of exponents in prime-power factorization of C(4n,n-1).

Original entry on oeis.org

0, 3, 3, 6, 4, 7, 8, 10, 9, 11, 8, 11, 10, 15, 14, 16, 13, 16, 14, 18, 15, 17, 17, 21, 20, 21, 19, 20, 18, 22, 21, 26, 21, 26, 24, 27, 23, 26, 28, 28, 28, 30, 26, 30, 27, 29, 29, 33, 30, 33, 30, 33, 30, 33, 31, 33, 31, 33, 32, 36, 35, 39, 39, 42, 37, 40, 38, 41, 40, 42, 40, 44, 41, 45, 44, 46, 44, 46

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001222, A004331, A022559, A023826.

Join[{0},Table[Total[Transpose[FactorInteger[Binomial[4n,n-1]]] [[2]]],{n,2,80}]] (* Harvey P. Dale, Jun 13 2013 *)
a[n_] := PrimeOmega[Binomial[4*n, n-1]]; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega(binomial(4*n, n-1)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A004331(n)). - Amiram Eldar, Jun 11 2025
a(n) = A022559(4*n) - A022559(3*n+1) - A022559(n-1).
a(n) = A023826(n) - A001222(3*n+1) + A001222(n). (End)

A004343 Binomial coefficient C(5n,n-1).

Original entry on oeis.org

1, 10, 105, 1140, 12650, 142506, 1623160, 18643560, 215553195, 2505433700, 29248649430, 342700125300, 4027810484880, 47465835030320, 560658857389200, 6635869816740560, 78682166288559225, 934433788613079150

Offset: 1

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 1..922
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001791, A004319, A004331, A004356, A004369, A004382.

From Peter Bala, Jul 21 2024: (Start)
a(n) = Sum_{k = 0..n-1} binomial(4*n+k, k).
a(n) = 5*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4)/((4*n - 4)*(4*n - 2)*(4*n - 1)*(4*n + 1)) * a(n-1) with a(1) = 1. (End)

A365340 a(n) = (4n)!/(3n+1)!.

Original entry on oeis.org

1, 1, 8, 132, 3360, 116280, 5100480, 271252800, 16963914240, 1220096908800, 99225500774400, 9003984596006400, 901928094049382400, 98856066097780992000, 11768525894839633920000, 1512185803617951221760000, 208598907329474462760960000

Offset: 0

Seiichi Manyama, Sep 01 2023

Cf. A001761, A001763, A052795, A365341.
Cf. A004331, A061924.
Cf. A370056, A370057, A370058.
Cf. A000142, A002293.

PARI
```
a(n) = (4*n)!/(3*n+1)!;
```

from sympy import ff
def A365340(n): return ff(n<<2,n-1) # Chai Wah Wu, Sep 01 2023

E.g.f.: exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * x^k/k ). - Seiichi Manyama, Feb 08 2024
a(n) = A000142(n)*A002293(n). - Alois P. Heinz, Feb 08 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^3).
a(n) = Sum_{k=0..n} (3*n+1)^(k-1) * |Stirling1(n,k)|. (End)

A257633 a(n) = binomial(4*n + 2,n).

Original entry on oeis.org

1, 6, 45, 364, 3060, 26334, 230230, 2035800, 18156204, 163011640, 1471442973, 13340783196, 121399651100, 1108176102180, 10142940735900, 93052749919920, 855420636763836, 7877932561061640, 72667580816130436, 671262558647881200, 6208770443303347920

Offset: 0

Peter Bala, Nov 04 2015

Michael De Vlieger, Table of n, a(n) for n = 0..1025
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Cf. A002293, A004331, A005810, A052203, A224274, A262977.

#A257633
seq(binomial(4*n + 2,n), n = 0..20);

Table[Binomial[4*n + 2, n], {n, 0, 120}] (* Michael De Vlieger, Apr 11 2025 *)

vector(30, n, n--; binomial(4*n+2, n)) \\ Altug Alkan, Nov 05 2015

The o.g.f. equals f(x)*g(x)^2, where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A052203 (k = 1), A224274 (k = 3) and A004331 (k = 4).

A004356 Binomial coefficient C(6n,n-1).

Original entry on oeis.org

1, 12, 153, 2024, 27405, 376992, 5245786, 73629072, 1040465790, 14783142660, 210980549208, 3022285436352, 43430966148115, 625806790696080, 9038619861406740, 130815226545884704, 1896724514424115530

Offset: 1

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 1..853
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001791, A004319, A004331, A004343, A004369, A004382.

Table[Binomial[6n,n-1],{n,20}] (* Harvey P. Dale, Mar 04 2017 *)

D-finite with recurrence 5*(n-1)*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) -72*(6*n-5)*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Mar 19 2025

cp's OEIS Frontend

A262977 a(n) = binomial(4*n-1,n).

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A224274 a(n) = binomial(4*n,n)/4.

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A004319 a(n) = binomial(3*n, n - 1).

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A052203 a(n) = (4n+1)*binomial(4n,n)/(3n+1).

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A348614 Numbers k such that the k-th composition in standard order has sum equal to twice its alternating sum.

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A023827 Sum of exponents in prime-power factorization of C(4n,n-1).

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A004343 Binomial coefficient C(5n,n-1).

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A365340 a(n) = (4n)!/(3n+1)!.