A037270 -id:A037270 - cp's OEIS frontend

A071253 a(n) = n^2*(n^2+1).

0, 2, 20, 90, 272, 650, 1332, 2450, 4160, 6642, 10100, 14762, 20880, 28730, 38612, 50850, 65792, 83810, 105300, 130682, 160400, 194922, 234740, 280370, 332352, 391250, 457652, 532170, 615440, 708122, 810900, 924482, 1049600, 1187010, 1337492, 1501850, 1680912

Offset: 0

N. J. A. Sloane, Jun 12 2002

The identity (n^5 + n^3)^2 + (n^2*(n^2 + 1))^2 = n*(n^3 + n)^3 can be written as A155977(n)^2 + a(n)^2 = n*A034262(n)^3. - Vincenzo Librandi, Aug 08 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A002522, A002378, A034262, A037270, A069187, A155977.

A071253:= func< n | 2*Binomial(n^2+1,2) >;
[A071253(n): n in [0..40]]; // G. C. Greubel, Sep 12 2024

with(combinat):seq(lcm(fibonacci(3,n),n^2),n=0..35); # Zerinvary Lajos, Apr 20 2008
a:=n->add(n+add(n+add(n, j=1..n-1),j=1..n),j=1..n):seq(a(n), n=0..41); # Zerinvary Lajos, Aug 27 2008

Table[(1/4) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n,0,40}] (* Artur Jasinski, Feb 10 2010 *)
Table[n^2*(n^2+1),{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[2 x (1+x) (1+4 x+x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

a(n)=n^2*(n^2+1) \\ Charles R Greathouse IV, Sep 24 2015

def A071253(n): return 2*binomial(n^2+1,2)
[A071253(n) for n in range(41)] # G. C. Greubel, Sep 12 2024

a(n) = A002522(n)*A000290(n). - Zerinvary Lajos, Apr 20 2008
a(n) = (1/4)*sinh(2*arcsinh(n))^2. - Artur Jasinski, Feb 10 2010
G.f.: 2*x*(1+x)*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Jan 08 2012
a(n) = A002378(A000290(n)). - Rick L. Shepherd, Sep 22 2014
Sum_{n>=1} 1/a(n) = 0.5682... = Pi^2/6- (Pi*coth Pi-1)/2 = A013661 - A259171 [J. Math. Anal. Appl. 316 (2006) 328]. - R. J. Mathar, Oct 18 2019
a(n) = 2*A037270(n). - R. J. Mathar, Oct 18 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - 1/2 + Pi*cosech(Pi)/2. - Amiram Eldar, Nov 05 2020
E.g.f.: exp(x)*x*(2 + 8*x + 6*x^2 + x^3). - Stefano Spezia, Oct 08 2022
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 16 2023

A173121 a(n) = sinh(2arccosh(n))^2 = 4n^2*(n^2 - 1).

Original entry on oeis.org

0, 0, 48, 288, 960, 2400, 5040, 9408, 16128, 25920, 39600, 58080, 82368, 113568, 152880, 201600, 261120, 332928, 418608, 519840, 638400, 776160, 935088, 1117248, 1324800, 1560000, 1825200, 2122848, 2455488, 2825760, 3236400, 3690240

Offset: 0

Artur Jasinski, Feb 10 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002415, A047928, A001079, A037270, A071253, A108741, A132592, A146311-A146313, A173115.

[4*n^2*(n^2-1): n in [0..40]]; // Vincenzo Librandi, Jun 15 2011

Table[4 n^2*(n^2 - 1), {n, 0, 30}] (* or *) Table[Round[N[Sinh[2 ArcCosh[n]]^2, 100]], {n, 0, 50}]
LinearRecurrence[{5,-10,10,-5,1},{0,0,48,288,960},40] (* Harvey P. Dale, Jul 22 2015 *)

a(n)=4*n^2*(n^2-1) \\ Charles R Greathouse IV, Jul 01 2013

a(n) = 48*A002415(n) = 4*A047928(n).
G.f.: 48*x^2*(1+x)/(1-x)^5. - Colin Barker, Mar 22 2012
From Amiram Eldar, Jul 26 2022: (Start)
Sum_{n>=2} 1/a(n) = (21 - 2*Pi^2)/48.
Sum_{n>=2} (-1)^n/a(n) = (Pi^2 - 9)/48. (End)

A173116 a(n) = sinh(2arcsinh(n))^2 = 4n^2*(n^2 + 1).

Original entry on oeis.org

0, 8, 80, 360, 1088, 2600, 5328, 9800, 16640, 26568, 40400, 59048, 83520, 114920, 154448, 203400, 263168, 335240, 421200, 522728, 641600, 779688, 938960, 1121480, 1329408, 1565000, 1830608, 2128680, 2461760, 2832488, 3243600

Offset: 0

Artur Jasinski, Feb 10 2010

			G.f. = 8*x + 80*x^2 + 360*x^3 + 1088*x^4 + 2600*x^5 + 5328*x^6 + 9800*x^7 + ... - _Michael Somos_, Jul 05 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173121.

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

Table[4*n^2*(n^2 + 1), {n, 0, 30}] (* OR *)
Table[Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 30}]
a[ n_] := TrigExpand @ Sinh[ 2 ArcSinh @ n]^2; (* Michael Somos, Jul 05 2018 *)

a(n)=4*n^2*(n^2+1) \\ Charles R Greathouse IV, Apr 17 2012

a(n)=8*binomial(n^2+1,2) \\ Charles R Greathouse IV, Apr 17 2012

a(n) = 4*A071253(n) = 8*A037270(n).
G.f.: 8*x*(1 + 5*x + 5*x^2 + x^3)/(1 - x)^5. - Colin Barker, Jan 08 2012
E.g.f.: 4*x*(2 + 8*x + 6*x^2 + x^3)*exp(x). - Michael Somos, Jul 05 2018
a(n) = a(-n) = (2*n)^2 + (2*n^2)^2 = (2*n^2 + 1)^2 - 1. - Michael Somos, Jul 05 2018
From Amiram Eldar, Oct 25 2024: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/24 + (1-Pi*coth(Pi))/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 + (Pi*cosech(Pi)-1)/8. (End)

Name corrected by Jianing Song, Nov 23 2018

A173129 a(n) = cosh(2 * n * arccosh(n)).

Original entry on oeis.org

1, 1, 97, 19601, 7380481, 4517251249, 4097989415521, 5170128475599457, 8661355881006882817, 18605234632923999244961, 49862414878754347585980001, 163104845048002042971670685041, 639582975902942936737758325440001

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..193
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A001079, A037270, A053120 (Chebyshev polynomial), A058331, A115066, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173148.

Cf. A349070, A349071, A349073.

seq(orthopoly[T](2*n,n), n=0..50); # Robert Israel, Dec 27 2018

Table[Round[Cosh[2 n ArcCosh[n]]], {n, 0, 20}] (* Artur Jasinski, Feb 10 2010 *)
Round[Table[1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x), {x, 0, 10}]] (* Artur Jasinski, Feb 14 2010 *)
Table[ChebyshevT[2*n, n], {n, 0, 15}] (* Vaclav Kotesovec, Nov 07 2021 *)

{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2-1)^(n-k)*n^(2*k))} \\ Seiichi Manyama, Dec 27 2018

{a(n) = polchebyshev(2*n, 1, n)} \\ Seiichi Manyama, Dec 28 2018

{a(n) = polchebyshev(n, 1, 2*n^2-1)} \\ Seiichi Manyama, Dec 29 2018

a(n) = (1/2)*((n+sqrt(n^2-1))^(2*n) + (n-sqrt(n^2-1))^(2*n)). - Artur Jasinski, Feb 14 2010, corrected by Vaclav Kotesovec, Apr 05 2016
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n^2-1)^(n-k)*n^(2*k). - Seiichi Manyama, Dec 27 2018
a(n) = T_{2n}(n) where T_{2n} is a Chebyshev polynomial of the first kind. - Robert Israel, Dec 27 2018
a(n) = T_{n}(2*n^2-1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018

A173127 a(n) = sinh((2n-1)*arcsinh(3)).

Original entry on oeis.org

-3, 3, 117, 4443, 168717, 6406803, 243289797, 9238605483, 350823718557, 13322062699683, 505887558869397, 19210405174337403, 729489509065951917, 27701390939331835443, 1051923366185543794917, 39945386524111332371403

Offset: 0

Artur Jasinski, Feb 10 2010

Numbers n such that ((n^2 + 1)/10) is a square. - Vincenzo Librandi, Jan 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (38,-1).

Cf. A058331 A001079, A037270, A071253, A108741, A132592, A146311-A146313, A173115, A173116, A173121.

[-3] cat [n: n in [0..10^7]|IsSquare((n^2+1)/10)]; // Vincenzo Librandi, Jan 02 2012

LinearRecurrence[{38,-1},{-3,3},30] (* Harvey P. Dale, Jan 14 2015 *)

from itertools import islice
def A173127_gen(): # generator of terms
    x, y = -30, 10
    while True:
        yield x//10
        x, y = x*19+y*60, x*6+y*19
A173127_list = list(islice(A173127_gen(),20)) # Chai Wah Wu, Apr 24 2025

a(n) = (1/2)*((-3+sqrt(10))*(19+6*sqrt(10))^n + (-3-sqrt(10))*(19-6*sqrt(10))^n).
a(n) = -a(-n+1).
G.f.: -3*(1-39*x)/(1-38*x+x^2). - Bruno Berselli, Jan 03 2011
E.g.f.: exp(19*x)*(-3*cosh(6*sqrt(10)*x) + sqrt(10)*sinh(6*sqrt(10)*x)). - Stefano Spezia, Apr 24 2025

A079547 a(n) = ((n^6 - (n-1)^6) - (n^2 - (n-1)^2))/60.

Original entry on oeis.org

0, 1, 11, 56, 192, 517, 1183, 2408, 4488, 7809, 12859, 20240, 30680, 45045, 64351, 89776, 122672, 164577, 217227, 282568, 362768, 460229, 577599, 717784, 883960, 1079585, 1308411

Offset: 1

Xavier Acloque, Jan 22 2003

Polynexus numbers of order 6.
A polynexus (subtractive) function is composed of two or more subtracted nexus numbers divided by an integer x. The general form of the formula is a(n)=((n^p-(n-1)^p)-(n^q-(n-1)^q))/x, where n, p, q and x are integers.
Already known: ((n^5-(n-1)^5) - (n^3-(n-1)^3))/24, giving A006322 for n>1; ((n^4-(n-1)^4) - (n^2-(n-1)^2))/12, giving A000330; ((n^3-(n-1)^3) - (n^1-(n-1)^1))/6, giving A000217; ((n^2-(n-1)^2) - (n^1-(n-1)^1))/2, giving n; ((n^2-(n-1)^2) - (n^0-(n-1)^0))/1, giving 2*n-1. In those examples, x is equal to 1,2,6,12,24, and 3 is also possible.
Also number of monotone n-weightings of complete bipartite digraph K(3,2) if offset were 0; cf. A085464-A085465. - Goran Kilibarda, Vladeta Jovovic, Jul 01 2003
Partial sums of A037270. - J. M. Bergot, Jun 07 2012

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006322, A000330, A000217, A047969, A003215, A083200, A088889-A088894.

List([1..30], n-> n*(6*n^4-15*n^3+20*n^2-15*n+4)/60) # G. C. Greubel, Jun 19 2019

[n*(6*n^4-15*n^3+20*n^2-15*n+4)/60: n in [1..30]]; // G. C. Greubel, Jun 19 2019

Table[((n^6 -(n-1)^6) - (n^2 -(n-1)^2))/60, {n, 30}] (* Bruno Berselli, Feb 13 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,11,56,192,517},30] (* Harvey P. Dale, Feb 21 2023 *)

a(n) = n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 \\ Charles R Greathouse IV, Jan 16 2013

[n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 for n in (1..30)] # G. C. Greubel, Jun 19 2019

a(n+1) = Sum_{i=1..n} (i^2 + i^4)/2 = n*(2*n+1)*(n+1)*(3*n^2+3*n+4)/60. - Vladeta Jovovic, Mar 17 2006
G.f.: x^2*(x+1)*(1+4*x+x^2)/(1-x)^6. - Bruno Berselli, Feb 13 2012
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^3. - Enrique Pérez Herrero, Jan 16 2013
E.g.f.: x^2*(30 + 80*x + 45*x^2 + 6*x^3)*exp(x)/60. - G. C. Greubel, Jun 19 2019

A173128 a(n) = cosh(2narcsinh(n)).

Original entry on oeis.org

1, 3, 161, 27379, 9478657, 5517751251, 4841332221601, 5964153172084899, 9814664424981012481, 20791777842234580902499, 55106605639755476546020001, 178627672869645203363556318483, 695165908550906808156689590141441

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Robert Israel, Table of n, a(n) for n = 0..192
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173129, A173174.

seq(expand( (1/2)*((n + sqrt(n^2 + 1))^(2*n) + (n - sqrt(n^2 + 1))^(2*n))), n=0..30); # Robert Israel, Apr 05 2016

Round[Table[Cosh[2 n ArcSinh[n]], {n, 0, 20}]] (* Artur Jasinski *)
Round[Table[1/2 (x - Sqrt[1 + x^2])^(2 x) + 1/2 (x + Sqrt[1 + x^2])^(2 x), {x, 0, 20}]] (* Artur Jasinski, Feb 14 2010 *)

{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2+1)^(n-k)*n^(2*k))} \\ Seiichi Manyama, Dec 27 2018

{a(n) = polchebyshev(n, 1, 2*n^2+1)} \\ Seiichi Manyama, Dec 29 2018

a(n) = (1/2)*((n + sqrt(n^2 + 1))^(2*n) + (n - sqrt(n^2 + 1))^(2*n)). - Artur Jasinski, Feb 14 2010, corrected by Vaclav Kotesovec, Apr 05 2016
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n^2+1)^(n-k)*n^(2*k). - Seiichi Manyama, Dec 27 2018
a(n) = T_{n}(2*n^2+1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018

A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.

Original entry on oeis.org

0, 1, 12, 51, 145, 330, 651, 1162, 1926, 3015, 4510, 6501, 9087, 12376, 16485, 21540, 27676, 35037, 43776, 54055, 66045, 79926, 95887, 114126, 134850, 158275, 184626, 214137, 247051, 283620, 324105, 368776, 417912, 471801, 530740, 595035, 665001, 740962

Offset: 0

Bruno Berselli, Jan 31 2014

After 0, first trisection of A011779 and right border of A177708.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A004188.
Cf. A000217, A000332, A011779, A177708.
Cf. similar sequences on the polygonal numbers: A002817(n) = A000217(A000217(n)); A000537(n) = A000290(A000217(n)); A037270(n) = A000217(A000290(n)); A062392(n) = A000384(A000217(n)).
Cf. sequences of the form A000217(m)+k*A000332(m+2): A062392 (k=12); A264854 (k=11); A264853 (k=10); this sequence (k=9); A006324 (k=8); A006323 (k=7); A000537 (k=6); A006322 (k=5); A006325 (k=4), A002817 (k=3), A006007 (k=2), A006522 (k=1).
Cf. A232713, A260810.

[n*(n+1)*(3*n^2+3*n-2)/8: n in [0..40]];

Table[n (n + 1) (3 n^2 + 3 n - 2)/8, {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,12,51,145},40] (* Harvey P. Dale, Aug 22 2016 *)

for(n=0, 40, print1(n*(n+1)*(3*n^2+3*n-2)/8", "));

G.f.: x*(1 + 7*x + x^2)/(1 - x)^5.
a(n) = a(-n-1) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A000326(A000217(n)).
a(n) = A000217(n) + 9*A000332(n+2).
Sum_{n>=1} 1/a(n) = 2 + 4*sqrt(3/11)*Pi*tan(sqrt(11/3)*Pi/2) = 1.11700627139319... . - Vaclav Kotesovec, Apr 27 2016

A327086 Array read by descending antidiagonals: A(n,k) is the number of achiral colorings of the edges of a regular n-dimensional simplex using up to k colors.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 10, 1, 5, 16, 45, 28, 1, 6, 25, 136, 387, 128, 1, 7, 36, 325, 2784, 8352, 792, 1, 8, 49, 666, 13125, 186304, 382563, 7620, 1, 9, 64, 1225, 46836, 2117750, 36507008, 44526672, 124344

Offset: 1

Robert A. Russell, Aug 19 2019

An n-dimensional simplex has n+1 vertices and (n+1)*n/2 edges. For n=1, the figure is a line segment with one edge. For n-2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with six edges. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. An achiral coloring is identical to its reflection.

A(n,k) is also the number of achiral colorings of (n-2)-dimensional regular simplices in an n-dimensional simplex using up to k colors. Thus, A(2,k) is also the number of achiral colorings of the vertices (0-dimensional simplices) of an equilateral triangle.

			Array begins with A(1,1):
  1  2   3    4     5     6      7      8      9      10      11      12 ...
  1  4   9   16    25    36     49     64     81     100     121     144 ...
  1 10  45  136   325   666   1225   2080   3321    5050    7381   10440 ...
  1 28 387 2784 13125 46836 137543 349952 797769 1667500 3248971 5973408 ...
  ...
For A(2,3) = 9, the colorings are AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, and CCC.

Robert A. Russell, Table of n, a(n) for n = 1..325 First 25 antidiagonals.
Harald Fripertinger, The cycle type of the induced action on 2-subsets
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

Cf. A327083 (oriented), A327084 (unoriented), A327085 (chiral), A327090 (exactly k colors), A325001 (vertices, facets), A337886 (faces, peaks), A337410 (orthotope edges, orthoplex ridges), A337414 (orthoplex edges, orthotope ridges).

Rows 1-4 are A000027, A000290, A037270, A331353.

CycleX[{2}] = {{1,1}}; (* cycle index for permutation with given cycle structure *)
CycleX[{n_Integer}] := CycleX[n] = If[EvenQ[n], {{n/2,1}, {n,(n-2)/2}}, {{n,(n-1)/2}}]
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i,1]] == s[[i-1,1]], s[[i-1,2]] += s[[i,2]]; s = Delete[s,i], Null]]; s)
CycleX[p_List] := CycleX[p] = compress[Join[CycleX[Drop[p, -1]], If[Last[p] > 1, CycleX[{Last[p]}], ## &[]], If[# == Last[p], {#, Last[p]}, {LCM[#, Last[p]], GCD[#, Last[p]]}] & /@ Drop[p, -1]]]
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[Total[If[EvenQ[Total[1-Mod[#,2]]], 0, pc[#] j^Total[CycleX[#]][[2]]] & /@ IntegerPartitions[n+1]]/((n+1)!/2)]
array[n_, k_] := row[n] /. j -> k
Table[array[n,d-n+1], {d,1,10}, {n,1,d}] // Flatten
(* Using Fripertinger's exponent per Andrew Howroyd's code in A063841: *)
pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))]
ex[v_] := Sum[GCD[v[[i]], v[[j]]], {i,2,Length[v]}, {j,i-1}] + Total[Quotient[v,2]]
array[n_,k_] := Total[If[OddQ[Total[1-Mod[#,2]]], pc[#]k^ex[#], 0] &/@ IntegerPartitions[n+1]]/((n+1)!/2)
Table[array[n,d-n+1], {d,10}, {n,d}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the vertices to a partition of n+1. It then determines the number of permutations for each partition and the cycle index for each partition.
A(n,k) = Sum_{j=1..(n+1)*n/2} A327090(n,j) * binomial(k,j).
A(n,k) = 2*A327084(n,k) - A327083(n,k) = A327083(n,k) - 2*A327085(n,k) = A327084(n,k) - A327085(n,k).

A282612 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to row permutations.

Original entry on oeis.org

0, 1, 120, 3654, 45760, 333375, 1703016, 6784540, 22500864, 64836045, 167167000, 393877506, 861456960, 1769830699, 3447273480, 6412923000, 11461636096, 19776716505, 33076889784, 53804808190, 85365336000, 132422893911, 201268229800, 300266132244, 440396812800

Offset: 0

David Nacin, Feb 19 2017

Cycle index of symmetry group is (3*s(2)^3*s(1)^3 + 2*s(3)^3 + s(1)^9)/6.

			The number of 3 X 3 binary matrices up to row permutations is 120.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A282613, A282614, A217331, A168555. A037270 (2x2 version.)

Mathematica
```
Table[(3n^6+2n^3+n^9)/6, {n, 0, 24}]
```

concat(0, Vec(x*(1 + 110*x + 2499*x^2 + 14500*x^3 + 26015*x^4 + 14934*x^5 + 2365*x^6 + 56*x^7) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

a(n) = n^3*(n^3+2)*(n+1)*(n^2-n+1)/6.

G.f.: x*(1 + 110*x + 2499*x^2 + 14500*x^3 + 26015*x^4 + 14934*x^5 + 2365*x^6 + 56*x^7) / (1 - x)^10. - Colin Barker, Feb 23 2017

cp's OEIS Frontend

A071253 a(n) = n^2*(n^2+1).

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A173121 a(n) = sinh(2*arccosh(n))^2 = 4*n^2*(n^2 - 1).

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A173116 a(n) = sinh(2*arcsinh(n))^2 = 4*n^2*(n^2 + 1).

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A173129 a(n) = cosh(2 * n * arccosh(n)).

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A173127 a(n) = sinh((2n-1)*arcsinh(3)).

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A079547 a(n) = ((n^6 - (n-1)^6) - (n^2 - (n-1)^2))/60.

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A173128 a(n) = cosh(2*n*arcsinh(n)).

A173121 a(n) = sinh(2arccosh(n))^2 = 4n^2*(n^2 - 1).

A173116 a(n) = sinh(2arcsinh(n))^2 = 4n^2*(n^2 + 1).

A173128 a(n) = cosh(2narcsinh(n)).

A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.