A004319 -id:A004319 - cp's OEIS frontend

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

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)

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

A004369 Binomial coefficient C(7n,n-1).

Original entry on oeis.org

1, 14, 210, 3276, 52360, 850668, 13983816, 231917400, 3872894697, 65033528560, 1096993404430, 18574174153080, 315502265971620, 5373846361969456, 91748617512913200, 1569699972909739440

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..804
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, A004356, A004382.

A023820 Sum of exponents in prime-power factorization of C(3n,n-1).

Original entry on oeis.org

0, 2, 4, 4, 4, 7, 7, 8, 8, 8, 11, 12, 9, 12, 13, 12, 11, 13, 14, 15, 14, 17, 18, 19, 16, 18, 20, 18, 18, 21, 20, 21, 19, 20, 22, 22, 19, 22, 24, 22, 21, 24, 27, 29, 28, 28, 29, 31, 27, 28, 29, 29, 28, 34, 32, 32, 31, 30, 34, 34, 32, 34, 36, 35, 31, 32, 31, 33, 33, 36, 35, 36, 32, 36, 40, 37, 36, 38

Offset: 1

Clark Kimberling

nonn

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

Cf. A001222, A004319, A023819, A023821, A023822, A023823, A023824, A023825.

Join[{0}, Table[Total[FactorInteger[Binomial[3 n, n - 1]][[All, 2]]], {n, 2, 78}]] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[3*n, n-1]]; Array[a, 100] (* Amiram Eldar, Jun 12 2025 *)

a(n) = bigomega(binomial(3*n, n-1)); \\ Amiram Eldar, Jun 12 2025

From Amiram Eldar, Jun 12 2025: (Start)
a(n) = A001222(A004319(n)).
a(n) = A023819(n) - A001222(2*n+1) + A001222(n). (End)

A120981 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 1 (n >= 0, k >= 0).

Original entry on oeis.org

1, 0, 3, 3, 0, 9, 1, 27, 0, 27, 18, 12, 162, 0, 81, 15, 270, 90, 810, 0, 243, 138, 270, 2430, 540, 3645, 0, 729, 189, 2898, 2835, 17010, 2835, 15309, 0, 2187, 1218, 4536, 34776, 22680, 102060, 13608, 61236, 0, 6561, 2280, 32886, 61236, 312984, 153090, 551124

Offset: 0

Emeric Deutsch, Jul 21 2006

A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.

			T(2,0)=3 because we have (Q,L,M), (Q,L,R) and (Q,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.
Triangle starts:
   1;
   0,   3;
   3,   0,   9;
   1,  27,   0,  27;
  18,  12, 162,   0, 81;
  15, 270,  90, 810,  0, 243;

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Diagonals include A129530, A036216.

Cf. A001764 (row sums), A004319, A120429, A120982, A120983, A120984.

T:=proc(n,k) if k<=n then (1/(n+1))*binomial(n+1,k)*sum(3^(3*j-n+2*k)*binomial(n+1-k,j)*binomial(j,n-k-2*j),j=0..n+1-k) else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[n_, k_] := (1/(n+1))*Binomial[n+1, k]*Sum[3^(2k - n + 3j)*Binomial[n + 1 - k, j]*Binomial[j, n - k - 2j], {j, 0, n - k + 1}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 02 2018 *)

T(n,k) = binomial(n+1, k)*sum(j=0, n+1-k, 3^(2*k-n+3*j)*binomial(n+1-k, j)*binomial(j, n-k-2*j))/(n+1); \\ Andrew Howroyd, Nov 06 2017

from sympy import binomial
def T(n, k): return binomial(n + 1, k)*sum([3**(2*k - n + 3*j)*binomial(n + 1 - k, j)*binomial(j, n - k - 2*j) for j in range(n + 2 - k)])//(n + 1)
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Nov 07 2017

T(n,0) = A120984(n).
Sum_{k>=1} k*T(n,k) = 3*binomial(3n,n-1) = 3*A004319(n).
T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..n+1-k} 3^(2k-n+3j)*binomial(n+1-k,j)*binomial(j,n-k-2j).
G.f.: G=G(t,z) satisfies G = 1 + 3tzG + 3z^2*G^2 + z^3*G^3.

A127040 a(n) = binomial(floor((3n+4)/2),floor(n/2)).

Original entry on oeis.org

1, 1, 5, 6, 28, 36, 165, 220, 1001, 1365, 6188, 8568, 38760, 54264, 245157, 346104, 1562275, 2220075, 10015005, 14307150, 64512240, 92561040, 417225900, 600805296, 2707475148, 3910797436, 17620076360, 25518731280, 114955808528

Offset: 0

T. D. Noe, Jan 03 2007

nonn

With offset 2, the number of compositions of n into floor(n/2) parts, which is an upper bound for A007874.

T. D. Noe, Table of n, a(n) for n=0..200

Cf. A004319 (bisection), A025174 (bisection), A099578.

seq(sum(binomial(n+k, k-1), k=0..ceil((n+1)/2)), n=0..28); # Zerinvary Lajos, Apr 11 2007

CoefficientList[Series[(-1 + (2 Cos[1/3 ArcSin[(3 Sqrt[3] x)/2]])/Sqrt[4 - 27 x^2] + 3 x^3 Hypergeometric2F1[4/3, 5/3, 5/2, (27 x^2)/4])/(3 x^2), {x, 0, 20}], x] (* Benedict W. J. Irwin, Aug 16 2016 *)
Table[Binomial[Floor[(3 n + 4)/2], Floor[n/2]], {n, 0, 28}] (* Michael De Vlieger, Aug 18 2016 *)

a(n) = binomial((3*n+4)\2, n\2); \\ Michel Marcus, Sep 09 2016

From Benedict W. J. Irwin, Aug 16 2016: (Start)
G.f.: (-1 + (2*cos(arcsin(3*sqrt(3)*x/2)/3))/sqrt(4-27*x^2) + 3*x^3*2F1(4/3,5/3;5/2;27*x^2/4))/(3*x^2).
E.g.f.: 2F3(4/3,5/3;1/2,3/2,2;27*x^2/16) + x*2F3(4/3,5/3;1,3/2,5/2;27*x^2/16).
(End)
D-finite with recurrence 8*(n+2)*(n+1)*a(n) -84*(n-1)*(n+1)*a(n-1) +6*(-33*n^2+54*n-8)*a(n-2) +9*(63*n^2-63*n-16)*a(n-3) +108*(3*n-5)*(3*n-7)*a(n-4)=0. - R. J. Mathar, Feb 08 2021

A100400 Triangle read by rows: T(n,k) is the number of nonroot nodes of outdegree k (0<=k<=n-1) in all non-crossing trees with n edges.

Original entry on oeis.org

1, 4, 2, 21, 12, 3, 120, 72, 24, 4, 715, 440, 165, 40, 5, 4368, 2730, 1092, 312, 60, 6, 27132, 17136, 7140, 2240, 525, 84, 7, 170544, 108528, 46512, 15504, 4080, 816, 112, 8, 1081575, 692208, 302841, 105336, 29925, 6840, 1197, 144, 9, 6906900, 4440150, 1973400, 708400, 212520, 53130, 10780, 1680, 180, 10

Offset: 1

Emeric Deutsch, Dec 30 2004

Row n contains n terms. Row sums yield A004319. Column 0 yields A045721.

			T(2,1)=2 because in the non-crossing trees /_, _\ and /\ we have 2 nonroot nodes of outdegree 1.
Triangle begins:
1;
4,2;
21,12,3;
120,72,24,4;

E. Deutsch and M. Noy, Statistics on non-crossing trees, Discrete Math., 254 (2002), 75-87.

Cf. A001764, A004319, A045721.

for n from 1 to 10 do seq((k+1)*binomial(3*n-k-2,2*n-1),k=0..n-1) od; # yields sequence in triangular form

T(n, k) = (k+1)binomial(3n-k-2, 2n-1) (0<=k<=n-1).

cp's OEIS Frontend

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

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

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A004369 Binomial coefficient C(7n,n-1).

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A023820 Sum of exponents in prime-power factorization of C(3n,n-1).

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A120981 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 1 (n >= 0, k >= 0).

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A127040 a(n) = binomial(floor((3n+4)/2),floor(n/2)).

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A100400 Triangle read by rows: T(n,k) is the number of nonroot nodes of outdegree k (0<=k<=n-1) in all non-crossing trees with n edges.

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