A062145 -id:A062145 - cp's OEIS frontend

A062143 Fifth column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

1, 40, 1080, 25200, 554400, 11975040, 259459200, 5708102400, 128432304000, 2968213248000, 70643475302400, 1733976211968000, 43927397369856000, 1148870392750080000, 31019500604252160000, 864410083505160192000

Offset: 0

Wolfdieter Lang, Jun 19 2001

The coefficients of the numerator polynomials N(m,x) of the e.g.f. for column m (here m=4) give triangle A062145.

			a(3) = (3+4)! * binomial(3+7,7) / 4! = (5040 * 120) / 24 = 25200. - _Indranil Ghosh_, Feb 23 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Index entries for sequences related to Laguerre polynomials

Cf. A062137, A062142, A062145.

[Factorial(n+4)*Binomial(n+7,7)/Factorial(4): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+4)!*Binomial[n+7,7]/4!,{n,0,15}] (* Indranil Ghosh, Feb 23 2017 *)

a(n) = (n+4)!*binomial(n+7,7)/4! \\ Indranil Ghosh, Feb 23 2017

import math
f=math.factorial
def C(n,r):return f(n)/f(r)/f(n-r)
def A062143(n):return f(n+4)*C(n+7,7)/f(4) # Indranil Ghosh, Feb 23 2017

a(n) = (n+4)!*binomial(n+7, 7)/4!;
E.g.f.: (1 + 28*x + 126*x^2 + 140*x^3 + 35*x^4)/(1-x)^12.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-4) = (-1)^n*f(n,4,-8), (n>=4). - Milan Janjic, Mar 01 2009

A105938 a(n) = binomial(n+2,2)*binomial(n+5,2).

Original entry on oeis.org

10, 45, 126, 280, 540, 945, 1540, 2376, 3510, 5005, 6930, 9360, 12376, 16065, 20520, 25840, 32130, 39501, 48070, 57960, 69300, 82225, 96876, 113400, 131950, 152685, 175770, 201376, 229680, 260865, 295120, 332640, 373626, 418285, 466830, 519480, 576460

Offset: 0

Zerinvary Lajos, Apr 27 2005

			If n=0 then C(0+2,0)*C(0+5,2) = C(2,0)*C(5,2) = 1*10 = 10.
If n=9 then C(9+2,9)*C(9+5,2) = C(11,9)*C(14,2) = 55*91 = 5005.

Nathaniel Johnston, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Subsequence of A085780.

Cf. A000096, A000217, A000389, A062145.

A105938:= func< n | 30*Binomial(n+5,5)/(n+3) >;
[A105938(n): n in [0..40]]; // G. C. Greubel, Mar 11 2025

a:= n-> binomial(n+2,n)*binomial(n+5,2):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 16 2008

Table[n(n+1)(n+3)(n+4)/4, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Table[Binomial[n + 2, n] Binomial[n + 5, 2], {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {10, 45, 126, 280, 540}, 40] (* Harvey P. Dale, Sep 05 2013 *)

def A105938(n): return 30*binomial(n+5,5)//(n+3)
print([A105938(n) for n in range(41)]) # G. C. Greubel, Mar 11 2025

G.f.: (10 - 5*x + x^2)/(1-x)^5. - Alois P. Heinz, Oct 16 2008
a(0)=10, a(1)=45, a(2)=126, a(3)=280, a(4)=540; for n>4, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Sep 05 2013
a(n) = A000217(n+1)*A000217(n+4). - R. J. Mathar, Nov 29 2015
a(n) = A000096(n+1)*A000096(n+2). - Bruno Berselli, Sep 21 2016
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=0} 1/a(n) = 5/36.
Sum_{n>=0} (-1)^n/a(n) = 1/12. (End)
From G. C. Greubel, Mar 11 2025: (Start)
a(n) = 30*A000389(n+5)/(n+3).
E.g.f.: (1/4)*(40 + 140*x + 92*x^2 + 18*x^3 + x^4)*exp(x). (End)

A105940 a(n) = binomial(n+5, 5)*binomial(n+8, 5).

Original entry on oeis.org

56, 756, 5292, 25872, 99792, 324324, 924924, 2378376, 5621616, 12388376, 25729704, 50791104, 95938752, 174350232, 306211752, 521694096, 864913896, 1399125420, 2213431220, 3431347920, 5221616400, 7811703900, 11504509380, 16698853080, 23914406880, 33821804016

Offset: 0

Zerinvary Lajos, Apr 27 2005

			a(0) = C(0+5,0)*C(0+8,5) = C(5,0)*C(8,5) = 1*56 = 56
a(6) = C(6+5,6)*C(6+8,5) = C(11,6)*C(14,5) = 462*2002 = 924924.

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A062145.

A105940:= func< n | Binomial(n+5,5)*Binomial(n+8,5) >;
[A105940(n): n in [0..40]]; // G. C. Greubel, Mar 11 2025

with(combinat); for i from 0 to 25 do print(i,numbcomb(i+5,i)*numbcomb(i+8,5)); end; # Jim Nastos, Oct 26 2005

a[n_] := Binomial[n + 5, 5] * Binomial[n + 8, 5]; Array[a, 25, 0] (* Amiram Eldar, Sep 01 2022 *)

def A105940(n): return binomial(n+5,5)*binomial(n+8,5)
print([A105940(n) for n in range(41)]) # G. C. Greubel, Mar 11 2025

G.f.: 28*(2+x)*(1+2*x) / (1-x)^11. - Colin Barker, Jan 28 2013
From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 580367/1764 - 100*Pi^2/3.
Sum_{n>=0} (-1)^n/a(n) = 74537/588 - 1280*log(2)/7. (End)

More terms from Jim Nastos, Oct 26 2005

A105942 a(n) = binomial(n+6,6)*binomial(n+9,6).

Original entry on oeis.org

84, 1470, 12936, 77616, 360360, 1387386, 4624620, 13741728, 37165128, 92912820, 217273056, 479693760, 1007356896, 2024399916, 3912705720, 7303717344, 13213962300, 23241027810, 39841761960, 66720654000, 109363854600, 175763337750, 277386503940, 430459323840

Offset: 0

Zerinvary Lajos, Apr 27 2005

			If n=0 then C(0+6,0)*C(0+9,6) = C(6,0)*C(9,6) = 1*84 = 84.
If n=6 then C(6+6,6)*C(6+9,6) = C(12,6)*C(15,6) = 924*5005 = 4624620.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A062145.

A105942:= func< n | Binomial(n+6,6)*Binomial(n+9,6)/42 >;
[A105942(n): n in [0..40]]; // G. C. Greubel, Mar 11 2025

Table[Binomial[n+6,n]Binomial[n+9,6],{n,0,30}] (* or *) CoefficientList[ Series[-((42 (x+1) (x (2 x+7)+2))/(x-1)^13),{x,0,30}],x] (* Harvey P. Dale, Sep 14 2012 *)

def A105942(n): return binomial(n+6,6)*binomial(n+9,6)
print([A105942(n) for n in range(41)]) # G. C. Greubel, Mar 11 2025

G.f.: 42*(1 + x)*(2 + 7*x + 2*x^2)/(1-x)^13. - Harvey P. Dale, Sep 14 2012
a(0)=84, a(1)=1470, a(2)=12936, a(3)=77616, a(4)=360360, a(5)=1387386, a(6)=4624620, a(7)=13741728, a(8)=37165128, a(9)=92912820, a(10)=217273056, a(11)=479693760, a(12)=1007356896, a(n) = 13*a(n-1) -78*a(n-2) +286*a(n-3) -715*a(n-4) +1287*a(n-5) -1716*a(n-6) +1716*a(n-7) -1287*a(n-8) +715*a(n-9) -286*a(n-10) +78*a(n-11) -13*a(n-12) +a(n-13). - Harvey P. Dale, Sep 14 2012
From Amiram Eldar, Sep 08 2022: (Start)
Sum_{n>=0} 1/a(n) = 10446039/3920 - 270*Pi^2.
Sum_{n>=0} (-1)^n/a(n) = 82911/560 - 15*Pi^2. (End)

Corrected and extended by Harvey P. Dale, Sep 14 2012

A105943 a(n) = binomial(n+7,7) * binomial(n+10,7).

Original entry on oeis.org

120, 2640, 28512, 205920, 1132560, 5096520, 19631040, 66745536, 204787440, 576438720, 1507608960, 3700494720, 8593371072, 19004570640, 40244973120, 81980500800, 161264274600, 307350735120, 569168028000, 1026681084000, 1807851474000, 3113521983000

Offset: 0

Zerinvary Lajos, Apr 27 2005

			If n=0 then C(0+7,0)*C(0+10,7) = C(7,0)*C(10,7) = 1*120 = 120.
If n=6 then C(6+7,6)*C(6+10,7) = C(13,6)*C(16,7) = 1716*11440 = 19631040.

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A062145.

A105943:= func< n | Binomial(n+7,7)*Binomial(n+10,7) >;
[A105943(n): n in [0..40]]; // G. C. Greubel, Mar 10 2025

A105943:=n->binomial(n+7,n)*binomial(n+10,7): seq(A105943(n), n=0..40); # Wesley Ivan Hurt, Apr 18 2017

Table[Binomial[n+7,n]Binomial[n+10,7],{n,0,30}] (* Harvey P. Dale, Nov 14 2011 *)

A105943_list, m = [], [3432, -3432, 1320, 0]+[120]*11
for _ in range(10**2):
    A105943_list.append(m[-1])
    for i in range(14):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

def A105943(n): return binomial(n+7,7)*binomial(n+10,7)
print([A105943(n) for n in range(41)]) # G. C. Greubel, Mar 10 2025

G.f.: 24*(5 + 35*x + 63*x^2 +35*x^3 + 5*x^4)/(1-x)^15. - Harvey P. Dale, Nov 14 2011
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 114905939/6480 - 5390*Pi^2/3.
Sum_{n>=0} (-1)^n/a(n) = 14336*log(2)/9 - 3577279/3240. (End)

More terms from Harvey P. Dale, Nov 14 2011

A105944 a(n) = binomial(n+8,n)*binomial(n+11,8).

Original entry on oeis.org

165, 4455, 57915, 495495, 3185325, 16563690, 73002930, 281582730, 972740340, 3062330700, 8904315420, 24168856140, 61764854580, 149660993790, 345855237750, 766005304350, 1632800780325, 3361648665375, 6705510829875, 12993932469375, 24518985616125

Offset: 0

Zerinvary Lajos, Apr 27 2005

			If n=0 then C(0+8,0)*C(0+11,8) = C(8,0)*C(11,8) = 1*165 = 165.
If n=4 then C(4+8,4)*C(4+11,8) = C(12,4)*C(15,8) = 495*6435 = 3185325.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. A030648, A062145.

A105944:= func< n | Binomial(n+8,8)*Binomial(n+11,8) >;
[A105944(n): n in [0..40]]; // G. C. Greubel, Mar 10 2025

Table[Binomial[n+8,n]Binomial[n+11,8],{n,0,30}] (* Harvey P. Dale, Apr 26 2018 *)

def A105994(n): return binomial(n+8,8)*binomial(n+11,8)
print([A105994(n) for n in range(41)]) # G. C. Greubel, Mar 10 2025

G.f.: 165*(1+x)*(1+9*x+19*x^2+9*x^3+x^4)/(1-x)^17. - Colin Barker, Jan 28 2013
From Amiram Eldar, Sep 08 2022: (Start)
Sum_{n>=0} 1/a(n) = 1493776559/14175 - 32032*Pi^2/3.
Sum_{n>=0} (-1)^n/a(n) = 112*Pi^2 - 1740989/1575. (End)
a(n) = 165*A030648(n). - G. C. Greubel, Mar 10 2025

More terms from Colin Barker, Jan 28 2013

A107417 a(n) = binomial(n+2,2)*binomial(n+5,5).

Original entry on oeis.org

1, 18, 126, 560, 1890, 5292, 12936, 28512, 57915, 110110, 198198, 340704, 563108, 899640, 1395360, 2108544, 3113397, 4503114, 6393310, 8925840, 12273030, 16642340, 22281480, 29484000, 38595375, 50019606, 64226358, 81758656, 103241160, 129389040, 161017472, 199051776

Offset: 0

Zerinvary Lajos, May 26 2005

			If n=0 then C(0+2,2)*C(0+5,5) = C(2,2)*C(5,5) = 1*1 = 1.
If n=3 then C(3+2,2)*C(3+5,5) = C(5,2)*C(8,5) = 10*56 = 560.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A062145.

A107417:= func< n | Binomial(n+2,n)*Binomial(n+5,n) >;
[A107417(n): n in [0..40]]; // G. C. Greubel, Mar 10 2025

Table[Binomial[n+2,2]Binomial[n+5,5],{n,0,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{1,18,126,560,1890,5292,12936,28512},40] (* Harvey P. Dale, Feb 18 2012 *)

for(n=0,40,print1(binomial(n+2,2)*binomial(n+5,5),","))

def A107417(n): return binomial(n+2,n)*binomial(n+5,n)
print([A107417(n) for n in range(41)]) # G. C. Greubel, Mar 10 2025

From Harvey P. Dale, Feb 18 2012: (Start)
a(0)=1, a(1)=18, a(2)=126, a(3)=560, a(4)=1890, a(5)=5292, a(6)=12936, a(7)=28512, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)- 28*a(n-6)+ 8*a(n-7)-a(n-8).
G.f.: (1 + 10*x + 10*x^2)/(1-x)^8. (End)
From Amiram Eldar, Sep 08 2022: (Start)
Sum_{n>=0} 1/a(n) = 25*Pi^2/3 - 5845/72.
Sum_{n>=0} (-1)^n/a(n) = 205/8 - 5*Pi^2/2. (End)
E.g.f.: (1/240)*(240 + 4080*x + 10920*x^2 + 9400*x^3 + 3350*x^4 + 542*x^5 + 39*x^6 + x^7)*exp(x). - G. C. Greubel, Mar 10 2025

More terms from Rick L. Shepherd, May 27 2005

A107419 a(n) = binomial(n+4,4)*binomial(n+7,7).

Original entry on oeis.org

1, 40, 540, 4200, 23100, 99792, 360360, 1132560, 3185325, 8179600, 19467448, 43439760, 91706160, 184497600, 355816800, 661028544, 1187785665, 2071432440, 3516320500, 5824819000, 9436206780, 14978106000, 23333661000, 35728290000, 53840548125, 79942445856

Offset: 0

Zerinvary Lajos, May 26 2005

			a(0) = C(0+4,4)*C(0+7,7) = C(4,4)*C(7,7) = 1*1 = 1.
a(6) = C(6+4,4)*C(6+7,7) = C(10,4)*C(13,7) = 210*1716 = 360360.

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A062145.

A107419:= func< n | Binomial(n+4,n)*Binomial(n+7,n) >;
[A107419(n): n in [0..40]]; // G. C. Greubel, Mar 10 2025

Table[Binomial[n+4,4]Binomial[n+7,7],{n,0,30}] (* Harvey P. Dale, Jul 27 2019 *)

for(n=0,29,print1(binomial(n+4,4)*binomial(n+7,7),","))

def A107419(n): return binomial(n+4,n)*binomial(n+7,n)
print([A107419(n) for n in range(41)]) # G. C. Greubel, Mar 10 2025

From Chai Wah Wu, Apr 10 2021: (Start)
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12) for n > 11.
G.f.: (1 + 28*x + 126*x^2 + 140*x^3 + 35*x^4)/(1 - x)^12. (End)
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 392*Pi^2 - 870268/225.
Sum_{n>=0} (-1)^n/a(n) = 56*Pi^2/3 - 3584*log(2)/15 - 441/25. (End)

Corrected and extended by Rick L. Shepherd, May 27 2005

A107420 a(n) = binomial(n+5,5)*binomial(n+8,8).

Original entry on oeis.org

1, 54, 945, 9240, 62370, 324324, 1387386, 5096520, 16563690, 48668620, 131405274, 330142176, 779502360, 1743502320, 3718285560, 7601828256, 14966099379, 28482196050, 52568991475, 94362067800, 165133618650, 282337298100, 472506635250, 775303893000, 1249100716500

Offset: 0

Zerinvary Lajos, May 26 2005

			a(0) = C(0+5,5)*C(0+8,8) = C(5,5)*C(8,8) = 1*1 = 1.
a(9) = C(9+5,5)*C(9+8,8) = C(14,5)*C(17,8) = 2002*24310 = 48668620.

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Cf. A062145.

A107420:= func< n | Binomial(n+5,n)*Binomial(n+8,n) >;
[A107420(n): n in [0..40]]; // G. C. Greubel, Mar 10 2025

a[n_] := Binomial[n + 5, 5] * Binomial[n + 8, 8]; Array[a, 30, 0] (* Amiram Eldar, Sep 06 2022 *)

for(n=0,29,print1(binomial(n+5,5)*binomial(n+8,8),","))

def A107420(n): return binomial(n+5,n)*binomial(n+8,n)
print([A107420(n) for n in range(41)]) # G. C. Greubel, Mar 10 2025

From Chai Wah Wu, Apr 10 2021: (Start)
a(n) = 14*a(n-1) - 91*a(n-2) + 364*a(n-3) - 1001*a(n-4) + 2002*a(n-5) - 3003*a(n-6) + 3432*a(n-7) - 3003*a(n-8) + 2002*a(n-9) - 1001*a(n-10) + 364*a(n-11) - 91*a(n-12) + 14*a(n-13) - a(n-14) for n > 13.
G.f.: (1 + 40*x + 280*x^2 + 560*x^3 + 350*x^4 + 56*x^5)/(1 - x)^14. (End)
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 2200*Pi^2 - 19150081/882.
Sum_{n>=0} (-1)^n/a(n) = 693421/490 - 20*Pi^2 - 12288*log(2)/7. (End)

More terms from Rick L. Shepherd, May 27 2005

A107421 a(n) = binomial(n+6,6)*binomial(n+9,9).

Original entry on oeis.org

1, 70, 1540, 18480, 150150, 924924, 4624620, 19631040, 73002930, 243343100, 739763024, 2078672960, 5456516520, 13495999440, 31674284400, 70950397056, 152432493675, 315413948850, 630827897700, 1223211990000, 2305754601150, 4235059471500, 7595106655500

Offset: 0

Zerinvary Lajos, May 26 2005

			If n=0 then C(0+6,6)*C(0+9,9) = C(6,6)*C(9,9) = 1*1 = 1.
If n=7 then C(7+6,6)*C(7+9,9) = C(13,6)*C(16,9) = 1716*11440 = 19631040.

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A062145.

A107421:= func< n | Binomial(n+6,n)*Binomial(n+9,n) >;
[A107421(n): n in [0..40]]; // G. C. Greubel, Mar 09 2025

Table[Binomial[n+6,6]Binomial[n+9,9],{n,0,30}] (* Harvey P. Dale, Jan 30 2013 *)

for(n=0,29,print1(binomial(n+6,6)*binomial(n+9,9),","))

def A107421(n): return binomial(n+6,n)*binomial(n+9,n)
print([A107421(n) for n in range(41)]) # G. C. Greubel, Mar 09 2025

G.f.: (1+54*x+540*x^2+1680*x^3+1890*x^4+756*x^5+84*x^6)/(1-x)^16. - Harvey P. Dale, Jan 30 2013
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 11583*Pi^2 - 4481289621/39200.
Sum_{n>=0} (-1)^n/a(n) = 73728*log(2)/35 - 225*Pi^2/2 - 13673259/39200. (End)

Corrected and extended by Rick L. Shepherd, May 27 2005

cp's OEIS Frontend

A062143 Fifth column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

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A105938 a(n) = binomial(n+2,2)*binomial(n+5,2).

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A105940 a(n) = binomial(n+5, 5)*binomial(n+8, 5).

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A105942 a(n) = binomial(n+6,6)*binomial(n+9,6).

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Magma

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SageMath

Formula

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A105943 a(n) = binomial(n+7,7) * binomial(n+10,7).

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Magma

Maple

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Python

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A105944 a(n) = binomial(n+8,n)*binomial(n+11,8).

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Magma

Mathematica