A132339 -id:A132339 - cp's OEIS frontend

A132341 Main diagonal of A132339.

1, 2, 10, 168, 4290, 136136, 4938024, 196125600, 8318177010, 370784099400, 17184867259560, 821870841735840, 40334204896057800, 2022686389717666848, 103312949950998743200, 5360873347802169267840, 282015983963437605168210

Offset: 0

N. J. A. Sloane, Nov 08 2007

G. C. Greubel, Table of n, a(n) for n = 0..500
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), see equation (67) circa p. 82.

Cf. A132339.

a[n_]:= If[n==0, 1, Binomial[2*n, n]*Binomial[4*n-2, 2*n-1]/(2*Binomial[2*n,2])];
Table[a[n], {n, 0, 20}] (* G. C. Greubel, Dec 14 2021 *)

a(n) = if (n, 2*(4*n-3)!/(n!^2*(2*n-1)!), 1); \\ Michel Marcus, Mar 27 2016

b=binomial
def a(n): return 1 if (n==0) else b(2*n, n)*b(4*n-2, 2*n-1)/(2*b(2*n,2))
[a(n) for n in (0..20)] # G. C. Greubel, Dec 14 2021

a(n) = T(n, n), where T(n,k) if the array of A132339.
a(n) = A(2*n, n), where A(n, k) is the antidiagonal triangle of A132339.
a(n) ~ 2^(6*n - 9/2) / (Pi*n^3). - Vaclav Kotesovec, Mar 27 2016
a(n) = binomial(2*n, n)*binomial(4*n-2, 2*n-1)/((2*n)*(2*n-1)), with a(0) = 1. - G. C. Greubel, Dec 14 2021

More terms from Max Alekseyev, Sep 12 2009

A006331 a(n) = n(n+1)(2*n+1)/3.

Original entry on oeis.org

0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770, 1012, 1300, 1638, 2030, 2480, 2992, 3570, 4218, 4940, 5740, 6622, 7590, 8648, 9800, 11050, 12402, 13860, 15428, 17110, 18910, 20832, 22880, 25058, 27370, 29820, 32412, 35150, 38038, 41080, 44280

Offset: 0

N. J. A. Sloane

Triangles in rhombic matchstick arrangement of side n.
Maximum accumulated number of electrons at energy level n. - Scott A. Brown, Feb 28 2000
Let M_n denote the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002
Convolution of odds (A005408) and evens (A005843). - Graeme McRae, Jun 06 2006
a(n) is the number of non-monotonic functions with domain {0,1,2} and codomain {0,1,...,n}. - Dennis P. Walsh, Apr 25 2011
For any odd number 2n+1, find Sum_{aJ. M. Bergot, Jul 16 2011
a(n) gives the number of (n+1) X (n+1) symmetric (0,1)-matrices containing three ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and |w - x| < y. - Clark Kimberling, Jun 02 2012
Partial sums of A001105. - Omar E. Pol, Jan 12 2013
Total number of square diagonals (of any size) in an n X n square grid. - Wesley Ivan Hurt, Mar 24 2015
Number of diagonal attacks of two queens on (n+1) X (n+1) chessboard. - Antal Pinter, Sep 20 2015
a(n) is the minimum value obtainable by partitioning either the set {x in the natural numbers | 1 <= x <= 2n} or the set {x in the natural numbers | 0 <= x <= 2n+1} into pairs, taking the product of all such pairs, and taking the sum of all such products. - Thomas Anton, Oct 21 2020
a(n) is the irregularity of the n-th power of a path of length at least 3*n.  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 16 2023
a(n) is the maximum possible total number of inversions in all rows and all columns of a Latin square of order n+1. - Ivaylo Kortezov, Jun 28 2025

			For n=2, a(2)=10 since there are 10 non-monotonic functions f from {0,1,2} to {0,1,2}, namely, functions f = <f(1),f(2),f(3)> given by <0,1,0>, <0,2,0>, <0,2,1>, <1,0,1>, <1,0,2>, <1,2,0>, <1,2,1>, <2,0,1>, <2,0,2>, and <2,1,2>. - _Dennis P. Walsh_, Apr 25 2011
Let n=4, 2*n+1 = 9. Since 9 = 1+8 = 3+6 = 5+4 = 7+2, a(4) = 1*8 + 3*6 + 5*4 + 7*2 = 60. - _Vladimir Shevelev_, May 11 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359.
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
Rowan Beckworth, Basic atomic information.
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 25.
N. S. S. Gu, H. Prodinger and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat., Vol. 31 (2010), pp. 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2 at n=3.
JBMO 2025, 29th Junior Balkan Mathematical Olympiad, Problem 4, author: Boris Mihov
Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, Vol. 6 (1965), circa p. 82.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Dennis Walsh, Notes on finite monotonic and non-monotonic functions.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A row of A132339.
Cf. A002378, A048147, A098077, A163102.
Cf. A005900, A084980, A077415, A112524, A188475, A100178, A035597, A096000.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

a006331 n = sum $ zipWith (*) [2*n-1, 2*n-3 .. 1] [2, 4 ..]
-- Reinhard Zumkeller, Feb 11 2012

[n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

A006331 := proc(n)
    n*(n+1)*(2*n+1)/3 ;
end proc:
seq(A006331(n),n=0..80) ; # R. J. Mathar, Sep 27 2013

Table[n(n+1)(2n+1)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,2,10,28},50] (* Harvey P. Dale, Apr 12 2013 *)

PARI
```
a(n)=if(n<0,0,n*(n+1)*(2*n+1)/3)
```

G.f.: 2*x*(1 + x)/(1 - x)^4. - Simon Plouffe (in his 1992 dissertation)
a(n) = 2*binomial(n+1,3) + 2*binomial(n+2,3).
a(n) = 2*A000330(n) = A002492(n)/2.
a(n) = Sum_{i=0..n} T(i,n-i), array T as in A048147. - N. J. A. Sloane, Dec 11 1999
From the formula for the sum of squares of positive integers 1^2 + 2^2 + 3^2 + ... + n^2 = n*(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum_{k=0..n} 2*k^2 = n*(n+1)*(2*n+1)/3, which is an alternative formula for this sequence. - Mike Warburton, Sep 08 2007
10*a(n) = A016755(n) - A001845(n); since A016755 are odd cubes and A001845 centered octahedral numbers, 10*a(n) are the "odd cubes without their octahedral contents." - Damien Pras, Mar 19 2011
a(n) = sum(a*b), where the summing is over all unordered partitions 2*n+1=a+b. - Vladimir Shevelev, May 11 2012
a(n) = binomial(2*n+2, 3)/2. - Ronan Flatley, Dec 13 2012
a(n) = A000292(n) + A002411(n). - Omar E. Pol, Jan 11 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, with a(0)=0, a(1)=2, a(2)=10, a(3)=28. - Harvey P. Dale, Apr 12 2013
a(n) = A208532(n+1,2). - Philippe Deléham, Dec 05 2013
Sum_{n>0} 1/a(n) = 9 - 12*log(2). - Enrique Pérez Herrero, Dec 03 2014
a(n) = A000292(n-1) + (n+1)*A000217(n). - J. M. Bergot, Sep 02 2015
a(n) = 2*(A000332(n+3) - A000332(n+1)). - Antal Pinter, Sep 20 2015
From Bruno Berselli, May 17 2018: (Start)
a(n) = n*A002378(n) - Sum_{k=0..n-1} A002378(k) for n>0, a(0)=0. Also:
A163102(n) = n*a(n) - Sum_{k=0..n-1} a(k) for n>0, A163102(0)=0. (End)
a(n) = A005900(n) - A000290(n) = A096000(n) - A000578(n+1) = A000578(n+1) - A084980(n+1) = A000578(n+1) - A077415(n)-1 = A112524(n) + 1 = A188475(n) - 1 = A061317(n) - A100178(n) = A035597(n+1) - A006331(n+1). - Bruce J. Nicholson, Jun 24 2018
E.g.f.: (1/3)*exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Jan 05 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi - 9. - Amiram Eldar, Jan 04 2022

A006332 From the enumeration of corners.

Original entry on oeis.org

0, 2, 28, 168, 660, 2002, 5096, 11424, 23256, 43890, 77924, 131560, 212940, 332514, 503440, 742016, 1068144, 1505826, 2083692, 2835560, 3801028, 5026098, 6563832, 8475040, 10829000, 13704210, 17189172, 21383208, 26397308, 32355010, 39393312, 47663616, 57332704

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006858, A132339.

[Binomial(n+2, 3)*Binomial(2*n+3, 3)/5: n in [0..30]]; // G. C. Greubel, Dec 14 2021

A006332:=-2*(1+z)*(z**2+6*z+1)/(z-1)**7; # conjectured by Simon Plouffe in his 1992 dissertation

Table[(n(1+n)^2(2+n)(1+2n)(3+2n))/90, {n, 0, 30}] (* or *)
{0}~Join~CoefficientList[Series[2(x+1)(x^2 +6x +1)/(1-x)^7, {x, 0, 29}], x] (* Michael De Vlieger, Mar 26 2016 *)

my(x='x+O('x^99)); concat(0, Vec(2*(x+1)*(x^2+6*x+1)/(1-x)^7)) \\ Altug Alkan, Mar 26 2016

[binomial(n+2, 3)*binomial(2*n+3, 3)/5 for n in (0..30)] # G. C. Greubel, Dec 14 2021

a(n) = (n*(1 + n)^2*(2 + n)*(1 + 2*n)*(3 + 2*n))/90.
a(n) = 2*A006858(n).
a(n) = (-1)^(n+1)*A132339(3, n).
G.f.: 2*(1+x)*(1 + 6*x + x^2)/(1-x)^7.
From G. C. Greubel, Dec 14 2021: (Start)
E.g.f.: (1/90)*x*(180 + 1080*x + 1350*x^2 + 555*x^3 + 84*x^4 + 4*x^5)*exp(x).
a(n) = binomial(n+2, 3)*binomial(2*n+3, 3)/5. (End)
From Amiram Eldar, Jul 10 2023: (Start)
Sum_{n>=1} 1/a(n) = 15*Pi^2 - 295/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = -15*Pi^2/2 + 120*Pi - 605/2. (End)

A006333 From the enumeration of corners.

Original entry on oeis.org

0, 2, 60, 660, 4290, 20020, 74256, 232560, 639540, 1586310, 3617900, 7696260, 15438150, 29451240, 53796160, 94607040, 160908264, 265670730, 427156860, 670609940, 1030350090, 1552346268, 2297341200, 3344614000, 4796473500

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Matthew House, Table of n, a(n) for n = 0..10000
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

A row of A132339.

Abs@ With[{n = 4}, Table[(2 (-1)^(n + k) (n + k - 1)! (2 n + 2 k - 3)!)/(n! k! (2 n - 1)! (2 k - 1)!), {k, 0, 24}]] (* or *)
{0}~Join~CoefficientList[Series[2 (1 + 20 x + 75 x^2 + 75 x^3 + 20 x^4 + x^5)/(1 - x)^10, {x, 0, 23}], x] (* Michael De Vlieger, Mar 26 2016 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,2,60,660,4290,20020,74256,232560,639540,1586310},30] (* Harvey P. Dale, Jan 01 2017 *)

a(n) = (n*(1 + n)^2*(2 + n)^2*(3 + n)*(1 + 2*n)*(3 + 2*n)*(5 + 2*n))/7560 \\ Charles R Greathouse IV, Jul 28 2015

x='x+O('x^99); concat(0, Vec(2*(1+20*x+75*x^2+75*x^3+20*x^4+x^5)/(1-x)^10)) \\ Altug Alkan, Mar 26 2016

a(n) = (n*(1 + n)^2*(2 + n)^2*(3 + n)*(1 + 2*n)*(3 + 2*n)*(5 + 2*n))/7560.

G.f.: 2*(1 + 20*x + 75*x^2 + 75*x^3 + 20*x^4 + x^5)/(1-x)^10.

A006334 From the enumeration of corners.

Original entry on oeis.org

0, 2, 110, 2002, 20020, 136136, 705432, 2984520, 10786908, 34370050, 98768670, 260390130, 638110200, 1468635168, 3200871520, 6650874912, 13248113736, 25415833170, 47143878782, 84832157410, 148507792972, 253549890440, 423093671000, 691331713800, 1107985378500

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

A row of A132339.

Abs@ With[{n = 5}, Table[(2 (-1)^(n + k) (n + k - 1)! (2 n + 2 k - 3)!)/(n! k! (2 n - 1)! (2 k - 1)!), {k, 0, 24}]] (* Michael De Vlieger, Mar 26 2016 *)
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,2,110,2002,20020,136136,705432,2984520,10786908,34370050,98768670,260390130,638110200},30] (* Harvey P. Dale, Apr 21 2016 *)

a(n) = (n*(1 + n)^2*(2 + n)^2*(3 + n)^2*(4 + n)*(1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n))/1360800.

G.f.: -2*x*(x+1)*(x^6 + 41*x^5 + 323*x^4 + 678*x^3 + 323*x^2 + 41*x + 1)/(x-1)^13. - Colin Barker, Sep 19 2012

cp's OEIS Frontend

A132341 Main diagonal of A132339.

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A006331 a(n) = n*(n+1)*(2*n+1)/3.

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A006332 From the enumeration of corners.

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Magma

Maple

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A006333 From the enumeration of corners.

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PARI

PARI

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A006334 From the enumeration of corners.

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Programs

Mathematica

Formula

A006331 a(n) = n(n+1)(2*n+1)/3.