A331436 -id:A331436 - cp's OEIS frontend

A002817 Doubly triangular numbers: a(n) = n(n+1)(n^2+n+2)/8.

0, 1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, 3081, 4186, 5565, 7260, 9316, 11781, 14706, 18145, 22155, 26796, 32131, 38226, 45150, 52975, 61776, 71631, 82621, 94830, 108345, 123256, 139656, 157641, 177310, 198765, 222111, 247456, 274911, 304590

Offset: 0

N. J. A. Sloane

Number of inequivalent ways to color vertices of a square using <= n colors, allowing rotations and reflections. Group is dihedral group D_8 of order 8 with cycle index (1/8)*(x1^4 + 2*x4 + 3*x2^2 + 2*x1^2*x2); setting all x_i = n gives the formula a(n) = (1/8)*(n^4 + 2*n + 3*n^2 + 2*n^3).
Number of semi-magic 3 X 3 squares with a line sum of n-1. That is, 3 X 3 matrices of nonnegative integers such that row sums and column sums are all equal to n-1. - [Gupta, 1968, page 653; Bell, 1970, page 279]. - Peter Bertok (peter(AT)bertok.com), Jan 12 2002. See A005045 for another version.
Also the coefficient h_2 of x^{n-3} in the shelling polynomial h(x)=h_0*x^n-1 + h_1*x^n-2 + h_2*x^n-3 + ... + h_n-1 for the independence complex of the cycle matroid of the complete graph K_n on n vertices (n>=2) - Woong Kook (andrewk(AT)math.uri.edu), Nov 01 2006
If X is an n-set and Y a fixed 3-subset of X then a(n-4) is equal to the number of 5-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Starting with offset 1 = binomial transform of [1, 5, 10, 9, 3, 0, 0, 0, ...]. - Gary W. Adamson, Aug 05 2009
Starting with "1" = row sums of triangle A178238. - Gary W. Adamson, May 23 2010
The equation n*(n+1)*(n^2 + n + 2)/8 may be arrived at by solving for x in the following equality: (n^2+n)/2 = (sqrt(8x+1)-1)/2. - William A. Tedeschi, Aug 18 2010
Partial sums of A006003. - Jeremy Gardiner, Jun 23 2013
Doubly triangular numbers are revealed in the sums of row sums of Floyd's triangle.
1, 1+5, 1+5+15, ...
             1
          2     3
       4     5     6
    7     8     9     10
11    12    13    14    15
- Tony Foster III, Nov 14 2015
From Jaroslav Krizek, Mar 04 2017: (Start)
For n>=1; a(n) = sum of the different sums of elements of all the nonempty subsets of the sets of numbers from 1 to n.
Example: for n = 6; nonempty subsets of the set of numbers from 1 to 3: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}; sums of elements of these subsets: 1, 2, 3, 3, 4, 5, 6; different sums of elements of these subsets: 1, 2, 3, 4, 5, 6; a(3) = (1+2+3+4+5+6) = 21, ... (End)
a(n) is also the number of 4-cycles in the (n+4)-path complement graph. - Eric W. Weisstein, Apr 11 2018

			G.f. = x + 6*x^2 + 21*x^3 + 55*x^4 + 120*x^5 + 231*x^6 + 406*x^7 + 666*x^8 + ...

A. Björner, The homology and shellability of matroids and geometric lattices, in Matroid Applications (ed. N. White), Encyclopedia of Mathematics and Its Applications, 40, Cambridge Univ. Press 1992.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(3,r).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics I, p. 292.

T. D. Noe and William A. Tedeschi, Table of n, a(n) for n = 0..10000 (first 1000 terms computed by T. D. Noe)
G. E. Andrews, P. Paule, A. Riese and V. Strehl, MacMahon's partition analysis V. Bijections, recursions and magic squares, p. 37.
Weymar Astaiza, Alexander J. Barrios, Henry Chimal-Dzul, Stephan Ramon Garcia, Jaaziel de la Luz, Victor H. Moll, Yunied Puig, and Diego Villamizar, Symmetric tensor powers of graphs, arXiv:2309.13741 [math.CO], 2023. See p. 12.
Matthias Beck, The number of "magic" squares and hypercubes, arXiv:math/0201013 [math.CO], 2002-2005.
A. G. Bell, Partitioning integers in n dimensions, The Computer Journal, 13 (1970), 278-283.
Miklos Bona, A New Proof of the Formula for the Number of 3 X 3 Magic Squares, Mathematics Magazine, Vol. 70, No. 3 (Jun., 1997), pp. 201-203.
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33(4) (1966), pp. 771-782.
Brian Conrey and Alex Gamburd, Pseudomoments of the Riemann zeta-function and pseudomagic squares, Journal of Number Theory, Volume 117, Issue 2, April 2006, Pages 263-278. See H4 on p. 269.
P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R2.
Robert W. Donley, Partitions for semi-magic squares of size three, arXiv:1911.00977 [math.CO], 2019.
I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Ann. Statist. 4 (1976), no. 6, 1159-1189.
I. J. Good, On the application of symmetric Dirichlet distributions and contingency tables, pp. 1178-1179. (Annotated scanned copy)
Hansraj Gupta, Enumeration of symmetric matrices, Duke Math. J. 35 (3), 653-659, (September 1968).
D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
Milan Janjic, Two Enumerative Functions
Neven Juric, Illustration of the 55 3 X 3 matrices
Michal Opler, Pavel Valtr, and Tung Anh Vu, On the Arrangement of Hyperplanes Determined by n Points, EuroCG (39th European Workshop on Computational Geometry, Barcelona, Spain 2023) Session 7B, Talk 1, Vol. 54, No. 6.
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
Henry Warburton, On Self-Repeating Series, Transactions of the Cambridge Philosophical Society, Vol. 9, 471-486, 1856.
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Path Complement Graph
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A006003 (first differences), A165211 (mod 2).
Multiple triangular: A000217, A064322, A066370.
Cf. A001496, A001621, A178238, A005045, A002721.
Cf. A006528 (square colorings).
Cf. A236770 (see crossrefs).
Cf. A016754, A007204.
Row n=3 of A257493 and row n=2 of A331436 and A343097.
Cf. A000332.
Cf. A000292 (3-cycle count of \bar P_{n+4}), A060446 (5-cycle count of \bar P_{n+3}), A302695 (6-cycle count of \bar P_{n+5}).

Maple
```
A002817 := n->n*(n+1)*(n^2+n+2)/8;
```

a[ n_] := n (n + 1) (n^2 + n + 2) / 8; (* Michael Somos, Jul 24 2002 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,1,6,21,55},40] (* Harvey P. Dale, Jul 18 2011 *)
nn=50;Join[{0},With[{c=(n(n+1))/2},Flatten[Table[Take[Accumulate[Range[ (nn(nn+1))/2]], {c,c}],{n,nn}]]]] (* Harvey P. Dale, Mar 19 2013 *)

{a(n) = n * (n+1) * (n^2 + n + 2) / 8}; /* Michael Somos, Jul 24 2002 */

concat(0, Vec(x*(1+x+x^2)/(1-x)^5 + O(x^50))) \\ Altug Alkan, Nov 15 2015

def A002817(n): return (m:=n*(n+1))*(m+2)>>3 # Chai Wah Wu, Aug 30 2024

a(n) = 3*binomial(n+2, 4) + binomial(n+1, 2).
G.f.: x*(1 + x + x^2)/(1-x)^5. - Simon Plouffe (in his 1992 dissertation); edited by N. J. A. Sloane, May 13 2008
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3. - Warut Roonguthai, Dec 13 1999
a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + a(n-5) = A000217(A000217(n)). - Ant King, Nov 18 2010
a(n) = Sum(Sum(1 + Sum(3*n))). - Xavier Acloque, Jan 21 2003
a(n) = A000332(n+1) + A000332(n+2) + A000332(n+3), with A000332(n) = binomial(n, 4). - Mitch Harris, Oct 17 2006 and Bruce J. Nicholson, Oct 22 2017
a(n) = Sum_{i=1..C(n,2)} i = C(C(n,2) + 1, 2) = A000217(A000217(n+1)). - Enrique Pérez Herrero, Jun 11 2012
Euler transform of length 3 sequence [6, 0, -1]. - Michael Somos, Nov 19 2015
E.g.f.: x*(8 + 16*x + 8*x^2 + x^3)*exp(x)/8. - Ilya Gutkovskiy, Apr 26 2016
Sum_{n>=1} 1/a(n) = 6 - 4*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) = 1.25269064911978447... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(n)*A000124(n)/2.
a(n) = ((n-1)^4 + 3*(n-1)^3 + 2*(n-1)^2 + 2*n))/8. - Bruce J. Nicholson, Apr 05 2017
a(n) = (A016754(n)+ A007204(n)- 2) / 32. - Bruce J. Nicholson, Apr 14 2017
a(n) = a(-1-n) for all n in Z. - Michael Somos, Apr 17 2017
a(n) = T(T(n)) where T are the triangular numbers A000217. - Albert Renshaw, Jan 05 2020
a(n) = 2*n^2 - n + 6*binomial(n, 3) + 3*binomial(n, 4). - Ryan Jean, Mar 20 2021
a(n) = (A008514(n) - 1)/16. - Charlie Marion, Dec 20 2024

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

A099121 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2}.

Original entry on oeis.org

1, 3, 21, 220, 3060, 53130, 1107568, 26978328, 752538150, 23667689815, 828931106355, 32006008361808, 1350990969850340, 61902409203193230, 3060335715568296000, 162392216278033616560, 9206887338937200407418

Offset: 0

Sascha Kurz, Sep 28 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 2, where the referees cannot be distinguished.

a(n) is the number of n element multisets of n element multisets of a 3-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=2 of A107862 and A331436.

Cf. A099122, A099123, A099124, A099125, A099126, A099127, A099128.

a(n)={binomial(binomial(n + 2, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial( (n+1)*(n+2)/2 + n-1, n).

a(n) = binomial(binomial(n + 2, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended by Andrew Howroyd, Jan 17 2020

A099122 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3}.

Original entry on oeis.org

1, 4, 55, 1540, 73815, 5461512, 581106988, 84431259000, 16104878212995, 3910294246315600, 1178924607035010836, 432472873725488656424, 189789513537655207705620, 98222259182333060014344720

Offset: 0

Sascha Kurz, Sep 28 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 3, where the referees cannot be distinguished.

a(n) is the number n element multisets of n element multisets of a 4-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=4 of A331436.

Cf. A099121, A099123, A099124, A099125, A099126, A099127, A099128.

a(n)={binomial(binomial(n+3, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n+3, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended by Andrew Howroyd, Jan 17 2020

A099123 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4}.

Original entry on oeis.org

1, 5, 120, 7770, 1088430, 286243776, 127860662755, 90079147136880, 94572327271677750, 141504997346476482290, 291098519807782284023426, 799388312264077003441393875, 2859142263297618955891805452700

Offset: 0

Sascha Kurz, Sep 28 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 4, where the referees cannot be distinguished.

a(n) is the number of n element multisets of n element multisets of a 5-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=5 of A331436.

Cf. A099121, A099122, A099124, A099125, A099126, A099127, A099128.

a(n)={binomial(binomial(n + 4, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n + 4, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended by Andrew Howroyd, Jan 17 2020

A099124 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5}.

Original entry on oeis.org

1, 6, 231, 30856, 11009376, 8809549056, 13949678575756, 39822612151165272, 190782296093487153627, 1449479533445348118223510, 16683660613067331275158983216, 280167196060745030529247396914000, 6651137552302201488023930244802896266

Offset: 0

Sascha Kurz, Sep 28 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 5, where the referees cannot be distinguished.

a(n) is the number of n element multisets of n element multisets of a 6-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=6 of A331436.

Cf. A099121, A099122, A099123, A099125, A099126, A099127, A099128.

Table[Binomial[Binomial[n+5,n]+n-1,n],{n,0,20}] (* Harvey P. Dale, Jul 26 2020 *)

a(n)={binomial(binomial(n + 5, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n + 5, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended and a(12) and beyond from Andrew Howroyd, Jan 17 2020

A099125 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6}.

Original entry on oeis.org

1, 7, 406, 102340, 83369265, 179224992408, 878487565272240, 8800321588119330984, 165564847349896309234920, 5470105884755875924791320090, 300550263698274781577833262263448, 26251679033395309424785182716562495776, 3509663406416043297299781592276029113718775

Offset: 0

Sascha Kurz, Sep 28 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 6, where the referees cannot be distinguished.

a(n) is the number of n element multisets of n element multisets of a 7-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=7 of A331436.

Cf. A099121, A099122, A099123, A099124, A099126, A099127, A099128.

a(n)={binomial(binomial(n + 6, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n + 6, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended and a(11) and beyond from Andrew Howroyd, Jan 17 2020

A099126 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6,7}.

Original entry on oeis.org

1, 8, 666, 295240, 503167995, 2629770332904, 35773664992355004, 1119582594247762626696, 73241437035618231162682185, 9277639855710782695858431981840, 2137918570337064383107929197622033920, 850936582591338109213109187016928388683280

Offset: 0

Sascha Kurz, Oct 11 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 7, where the referees cannot be distinguished.

a(n) is the number of n element multisets of n element multisets of an 8-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=8 of A331436.

Cf. A099121, A099122, A099123, A099124, A099125, A099127, A099128.

a(n)={binomial(binomial(n + 7, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n + 7, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended and a(11) and beyond from Andrew Howroyd, Jan 17 2020

A099127 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6,7,8}.

Original entry on oeis.org

1, 9, 1035, 762355, 2531986380, 29653914688398, 1023687680214527328, 90954904732217610881940, 18709083803797153776767847375, 8183604949527627465377060678018870, 7099997495119970047949715137555520213198

Offset: 0

Sascha Kurz, Oct 11 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 8, where the referees cannot be distinguished.

a(n) is the number of n element multisets of n element multisets of a 9-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k=9 of A331436.

Cf. A099121, A099122, A099123, A099124, A099125, A099126, A099128.

a(n)={binomial(binomial(n + 8, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n + 8, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended and a(10) and beyond from Andrew Howroyd, Jan 17 2020

A099128 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6,7,8,9}.

Original entry on oeis.org

1, 10, 1540, 1798940, 10981240985, 269343686017406, 21897427636095471460, 5097399860176368033512080, 3028721298862926523085514684685, 4186904993091626163441378607213473000, 12477686558866630120430437118910496237274716

Offset: 0

Sascha Kurz, Oct 11 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 9, where the referees cannot be distinguished.

a(n) is the number of n element multisets of n element multisets of a 10-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k=10 of A331436.

Cf. A099121, A099122, A099123, A099124, A099125, A099126, A099127.

a(n)={binomial(binomial(n + 9, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n + 9, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended and a(10) and beyond from Andrew Howroyd, Jan 17 2020

A140236 a(n) = A000292(A000292(n)).

Original entry on oeis.org

0, 1, 20, 220, 1540, 7770, 30856, 102340, 295240, 762355, 1798940, 3939936, 8104460, 15803060, 29426320, 52636760, 90889616, 152112005, 247574180, 392991060, 609896980, 927341646, 1383960600, 2030479100, 2932714200, 4175145975, 5865135276, 8137872120

Offset: 0

Artur Jasinski, May 14 2008

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Row n=3 of A331436.

Cf. A000292.

a:= (n-> binomial(n+2,3))@@2:
seq(a(n), n=0..29);  # Alois P. Heinz, Mar 11 2024

Table[(n (1 + n) (2 + n) (3 + n) (2 + n^2) (12 + n (1 + n) (2 + n)))/1296,{n,0,20}]

b(n)=n*(n+1)*(n+2)/6;
a(n)=b(b(n));
vector(25,n,a(n-1)) \\ Joerg Arndt, Mar 11 2024

a(n) = (n*(1 + n)*(2 + n)*(3 + n)*(2 + n^2)*(12 + n*(1 + n)*(2 + n)))/1296.

G.f.: x*(1+10*x+65*x^2+120*x^3+80*x^4+4*x^5)/(1-x)^10. - Colin Barker, Apr 30 2012.

cp's OEIS Frontend

A002817 Doubly triangular numbers: a(n) = n*(n+1)*(n^2+n+2)/8.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A099121 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A099122 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A099123 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A099124 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A099125 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A099126 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6,7}.

Views

Author

Keywords

Comments

Links

A002817 Doubly triangular numbers: a(n) = n(n+1)(n^2+n+2)/8.