A002694 -id:A002694 - cp's OEIS frontend

A232224 Number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 3 simple intersections.

0, 0, 0, 1, 20, 195, 1430, 9009, 51688, 278460, 1434120, 7141530, 34648856, 164663785, 769491450, 3546222225, 16152872400, 72846725160, 325722299760, 1445598337950, 6373942543800, 27942072562950, 121863923024844, 529043313674106, 2287209524819120

Offset: 0

N. J. A. Sloane, Nov 22 2013

nonn

Lars Blomberg, Table of n, a(n) for n = 0..100
V. Pilaud, J. Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv preprint arXiv:1307.6440 [math.CO], 2013.

Cf. A002694, A074922, A274404.

CoefficientList[Series[(1 - Sqrt[1 - 4 x^2])^6 ((1 - x^2) Sqrt[1 - 4 x^2] + 7 x^2 - 26 x^4)/(64 x^6 Sqrt[1 - 4 x^2]^5), {x, 0, 48}], x^2] (* Michael De Vlieger, Sep 30 2015 *)

lista(nn) = {np = 2*nn+2; default(seriesprecision, np); pol = (1-sqrt(1-4*x^2))^6*((1-x^2)*sqrt(1-4*x^2)+7*x^2-26*x^4)/(64*x^6*sqrt(1-4*x^2)^5) + O(x^(np)); forstep (n=0, 2*nn, 2, print1(polcoeff(pol, n), ", "););} \\ Michel Marcus, Sep 30 2015

x='x+O('x^33); concat([0,0,0],Vec((1-sqrt(1-4*x))^6*((1-x)*sqrt(1-4*x)+7*x-26*x^2) / (64*x^3*sqrt(1-4*x)^5))) \\ Joerg Arndt, Sep 30 2015

Pilaud-Rue give an explicit g.f.

a(n) = [x^(2n)] (1-sqrt(1-4*x^2))^6*((1-x^2)*sqrt(1-4*x^2)+7*x^2-26*x^4) / (64*x^6*sqrt(1-4*x^2)^5). - Michel Marcus, Sep 30 2015

Corrected initial terms and more terms from Lars Blomberg, Sep 30 2015

A281580 a(n) = binomial(9*n, n-9).

Original entry on oeis.org

1, 90, 4851, 204156, 7413705, 244222650, 7511839335, 219683466288, 6183023199255, 168899639028120, 4505395859893071, 117891537949758600, 3036500678480436531, 77190387796530738576, 1940723247304668029175, 48339506285032758609456, 1194448077521704400002650

Offset: 9

Vincenzo Librandi, Feb 02 2017

Row 9*n, column n-9 of A007318. - Felix Fröhlich, Feb 05 2017

Cf. sequences with formula binomial(k*n, n-k): A002694 (k=2), A004321 (k=3), A004334 (k=4), A004347 (k=5), A004361 (k=6), A004375 (k=7), A004389 (k=8), this sequence (k=9).

Magma
```
[Binomial(9*n, n-9): n in [9..30]];
```
Mathematica
```
Table[Binomial[9 n, n - 9], {n, 9, 25}]
```

a(n) = binomial(9*n, n-9) \\ Felix Fröhlich, Feb 05 2017

A023818 Sum of exponents in prime-power factorization of C(2n,n-2).

Original entry on oeis.org

0, 2, 3, 5, 4, 4, 6, 8, 6, 7, 8, 9, 8, 10, 11, 12, 11, 11, 12, 14, 11, 12, 14, 15, 13, 15, 16, 16, 13, 13, 17, 19, 14, 16, 18, 17, 15, 18, 19, 22, 20, 20, 21, 24, 21, 20, 23, 24, 23, 24, 22, 23, 22, 23, 24, 25, 22, 24, 27, 27, 22, 26, 29, 30, 26, 26, 28, 30, 27, 28, 31, 31, 29, 31, 31, 33, 31, 28, 31

Offset: 2

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A001222, A002694, A023816, A023817.

Join[{0}, Table[Total[FactorInteger[Binomial[2 n, n - 2]][[All, 2]]], {n, 3, 80}]] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[2*n, n-2]]; Array[a, 100, 2] (* Amiram Eldar, Jun 12 2025 *)

a(n) = bigomega(binomial(2*n, n-2)); \\ Amiram Eldar, Jun 12 2025

From Amiram Eldar, Jun 12 2025: (Start)
a(n) = A001222(A002694(n)).
a(n) = A023817(n) - A022559(n+2) + A022559(n-1). (End)

Offset corrected to 2 by Ivan Neretin, Nov 02 2017

A026389 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=4; also a(n) = T(2n,n-2).

Original entry on oeis.org

1, 8, 49, 272, 1440, 7428, 37730, 189808, 948909, 4724160, 23453001, 116207424, 575036475, 2842936320, 14046869575, 69378730880, 342590699955, 1691519468760, 8351553940355, 41235710124400, 203617691311370, 1005560117623204

Offset: 2

Clark Kimberling

nonn

Binomial transform of A002694. - Ross La Haye, Mar 05 2005

Conjecture: (n+2)*a(n) +4*(-3*n-2)*a(n-1) +2*(24*n-19)*a(n-2) +4*(-18*n+43)*a(n-3) +35*(n-4)*a(n-4)=0. - R. J. Mathar, May 29 2013

A092583 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-pattern is equal to k.

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 4, 6, 14, 0, 20, 30, 28, 42, 0, 120, 180, 168, 120, 132, 0, 840, 1260, 1176, 840, 495, 429, 0, 6720, 10080, 9408, 6720, 3960, 2002, 1430, 0, 60480, 90720, 84672, 60480, 35640, 18018, 8008, 4862, 0, 604800, 907200, 846720, 604800, 356400, 180180, 80080, 31824, 16796

Offset: 1

Emeric Deutsch and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004

Row sums are the factorial numbers (A000142).
Diagonal is A000108.
T(n,n-1) = binomial(2n-2,n-3) = A002694(n-1).

			T(4,3) = 6 because 1324, 1423, 2134, 2314, 3124 and 4123 are the only permutations of [4] in which the length of the longest initial segment avoiding the 123-pattern is equal to 3 (i.e., the first three entries do not contain the 123-pattern but all 4 of them do).
Triangle starts:
  1;
  0,    2;
  0,    1,    5;
  0,    4,    6,   14;
  0,   20,   30,   28,   42;
  0,  120,  180,  168,  120,  132;
  0,  840, 1260, 1176,  840,  495,  429;
  ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened
E. Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.
R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985.

Cf. A000142, A000108, A002694.

T:= function(n,k)
    if k=n then return Binomial(2*n, n)/(n + 1);
    else return Factorial(n)*Binomial(2*k, k-2)/Factorial(k+1);
    fi;
  end;
Flat(List([1..12], n-> List([1..n], k-> T(n,k) ))); # G. C. Greubel, Jul 22 2019

T:= func< n,k | k eq n select Catalan(n) else Factorial(n)*Binomial(2*k, k-2)/Factorial(k+1) >;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 22 2019

T[n_,k_]:= If[k==n, CatalanNumber[n], n!*Binomial[2*k,k-2]/(k+1)!]; Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 22 2019 *)

tabl(nn) = {for (n=1, nn, for (k=1, n-1, print1(n!*binomial(2*k, k-2)/(k+1)!, ", ");); print1(binomial(2*n, n)/(n+1), ", "); print(););} \\ Michel Marcus, Jul 16 2013

def T(n, k):
    if (k==n): return catalan_number(n)
    else: return factorial(n)*binomial(2*k, k-2)/factorial(k+1)
[[T(n,k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 22 2019

T(n,k) = n!*binomial(2k, k-2)/(k+1)! for k < n;

T(n,n) = binomial(2n, n)/(n+1) = A000108(n).

A113857 a(n) = binomial(4+2n, n) binomial(9+2*n, 4+n).

Original entry on oeis.org

126, 2772, 48048, 772200, 12033450, 184940756, 2824549728, 43028530272, 655081791000, 9977399586000, 152112583402560, 2322021633001200, 35496198345658050, 543418421128852500, 8331507823355640000, 127919340117331963200, 1966759854303978934200, 30279186980267369086800

Offset: 0

Zerinvary Lajos, Feb 02 2006

If one uses the "table" view of array A062190, the sequence appears as the fourth column right from the middle in the "formatted as a triangular array" subpanel.

			a(0) = C(4+2*n,n)*C(9+2*n,4+n) = C(4,0)*C(9,4) = 1*126 = 126.
a(7) = C(4+2*7,7)*C(9+2*7,4+7) = C(18,7)*C(23,11) = 31824*1352078 = 43028530272.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A001700, A002694, A062190.

a[n_] := Binomial[4+2*n, n] * Binomial[9+2*n, 4+n]; Array[a, 20, 0] (* Amiram Eldar, Sep 05 2025 *)

a(n)={binomial(4+2*n, n) * binomial(9+2*n, 4+n)} \\ Andrew Howroyd, Jan 07 2020

a(n) = A062190(4+2*n, 4+n).
a(n) = A002694(n+2)*A001700(n+4). - R. J. Mathar, Nov 28 2008
a(n) ~ 2^(4*n+13) / (Pi*n). - Amiram Eldar, Sep 05 2025

Definition rephrased by R. J. Mathar, Nov 28 2008

Edited and more terms added by Andrew Howroyd, Jan 07 2020

A141134 Hankel transform of C(2n+4,n+4).

Original entry on oeis.org

1, -8, 8, 1, 1, -16, 16, 1, 1, -24, 24, 1, 1, -32, 32, 1, 1, -40, 40, 1, 1, -48, 48, 1, 1, -56, 56, 1, 1, -64, 64, 1, 1, -72, 72, 1, 1, -80, 80, 1, 1, -88, 88, 1, 1, -96, 96, 1, 1, -104, 104

Offset: 0

Paul Barry, Jun 06 2008

Hankel transform of A002694(n+2).

Hankel transform of A002694(n+1) is sin(Pi*n/2)*(cos(Pi*n)/2-1/2).

Index entries for linear recurrences with constant coefficients, signature (-2,-3,-4,-3,-2,-1).

Cf. A002694.

LinearRecurrence[{-2,-3,-4,-3,-2,-1},{1,-8,8,1,1,-16},60] (* Harvey P. Dale, Jun 22 2025 *)

G.f.: ( 1-6*x-5*x^2-3*x^3-2*x^4-x^5 ) / ( (1+x)^2*(x^2+1)^2 ).

A085391 Square array of centered numbers, read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 4, 5, 1, 0, 1, 5, 10, 7, 1, 0, 1, 6, 15, 19, 9, 1, 0, 1, 7, 21, 35, 31, 11, 1, 0, 1, 8, 28, 56, 69, 46, 13, 1, 0, 1, 9, 36, 84, 126, 121, 64, 15, 1, 0, 1, 10, 45, 120, 210, 251, 195, 85, 17, 1, 0, 1, 11, 55, 165, 330, 462, 456, 295, 109, 19, 1, 0

Offset: 0

Paul Barry, Jul 02 2003

			Rows begin
0 0 0 0 0 0 ...
1 1 1 1 1 1 ...
1 3 5 7 9 11 ...
1 4 10 19 31 46 ...
1 5 15 35 69 121...

Rows include A005408, A005448, A005894, A008498, A008499, A008500, A008501.

Diagonals include A030662, A001700, A001791, A002054, A002694, A003516.

Square array T(n, k)=C(n+k, k)-C(n, k).

Row k has g.f. (1-x^k)/(1-x)^(k+1).

A090985 Triangle read by rows: T(n,k) is the number of dissections of a convex n-gon by nonintersecting diagonals, having exactly k triangles (n >= 2, k >= 0).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 5, 0, 5, 4, 6, 21, 0, 14, 8, 35, 28, 84, 0, 42, 25, 80, 216, 120, 330, 0, 132, 64, 309, 540, 1155, 495, 1287, 0, 429, 191, 890, 2475, 3080, 5720, 2002, 5005, 0, 1430, 540, 3058, 7788, 16302, 16016, 27027, 8008, 19448, 0, 4862, 1616, 9580, 30108, 54964, 96005, 78624, 123760, 31824, 75582, 0, 16796

Offset: 2

Emeric Deutsch, Feb 28 2004

T(n,n-2) = [binomial(2n-4, n-2)]/(n-1) = Catalan(n-2) (A000108).
T(n,n-4) = binomial(2n-5, n-4) (A002054).
T(n,n-5) = binomial(2n-6, n-5) (A002694).
T(n,0) = A046736(n).
Row sums give the little Schroeder numbers (A001003).

			T(5,1)=5 because the dissections of a convex pentagon having exactly one triangle are obtained by the placement of a diagonal between any pair of non-adjacent vertices.
T(6,0)=4 because the dissections of a convex hexagon with no triangles are obtained by the null placement and by placing one diagonal between any of the 3 pairs of opposite vertices.
Triangle starts:
  1;
  0,  1;
  1,  0,  2;
  1,  5,  0,  5;
  4,  6, 21,  0, 14;
  8, 35, 28, 84,  0, 42;
  ...

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

Cf. A000108, A002054, A002694, A046736.

T := (n,k)->binomial(n+k-2,k)*sum(binomial(n-2+k+i,i)*binomial(n-3-k-i,i-1), i=0..floor((n-2-k)/2))/(n-1): seq(seq(T(n,k),k=0..n-2),n=2..14);

T [n_, k_] := Binomial[n+k-2, k] Sum[Binomial[n-2+k+i, i] Binomial[n-3-k-i, i-1], {i, 0, (n-2-k)/2}]/(n-1);
Table[T[n, k], {n, 2, 12}, {k, 0, n-2}] // Flatten (* Jean-François Alcover, Jul 29 2018 *)

T(n, k) = binomial(n+k-2, k)*(Sum_{i=0..floor((n-2-k)/2)} binomial(n-2+k+i, i)*binomial(n-3-k-i, i-1))/(n-1).

G.f.: G=G(t, z) satisfies (1-t)G^3 + (1+t)zG^2 - z^2*(1+z)G + z^4 = 0.

A113888 a(n) = C(2n+1,n)C(2*n+6,n+1).

Original entry on oeis.org

6, 84, 1200, 17325, 252252, 3699696, 54609984, 810616950, 12092280200, 181176906768, 2725140250560, 41132585656890, 622787147955000, 9456196695480000, 143946539451475200, 2196309308974461450, 33581927605139911800, 514470608092210770000, 7895695609776494520000

Offset: 0

Zerinvary Lajos, Jan 28 2006

			If n=0 then C(1+2*0,0+0)*C(6+2*0,1+0) = C(1,0)*C(6,1) = 1*6 = 6.
If n=4 then C(1+2*4,0+4)*C(6+2*4,1+4) = C(9,4)*C(14,5) = 126*2002 = 252252.
If n=10 then C(1+2*10,0+10)*C(6+2*10,1+10) = C(21,10)*C(26,11) = 352716*7726160 = 2725140250560.

Cf. A001700, A002694, A062190.

Table[Binomial[2n+1,n]Binomial[2n+6,n+1],{n,0,20}] (* Harvey P. Dale, Jun 14 2011 *)

From Amiram Eldar, Sep 06 2025: (Start)
a(n) = A001700(n) * A002694(n+3).
a(n) ~ 2^(4*n+7) / (Pi*n). (End)

Edited by N. J. A. Sloane, Feb 03 2007

cp's OEIS Frontend

A232224 Number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 3 simple intersections.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A281580 a(n) = binomial(9*n, n-9).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

PARI

A023818 Sum of exponents in prime-power factorization of C(2n,n-2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A026389 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=4; also a(n) = T(2n,n-2).

Views

Author

Keywords

Comments

Formula

A092583 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-pattern is equal to k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A113857 a(n) = binomial(4+2*n, n) * binomial(9+2*n, 4+n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A141134 Hankel transform of C(2n+4,n+4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A085391 Square array of centered numbers, read by antidiagonals.

Views

A113857 a(n) = binomial(4+2n, n) binomial(9+2*n, 4+n).

A113888 a(n) = C(2n+1,n)C(2*n+6,n+1).