A036289 -id:A036289 - cp's OEIS frontend

A128964 a(n) = (n^3-n)*6^n.

0, 216, 5184, 77760, 933120, 9797760, 94058496, 846526464, 7255941120, 59861514240, 478892113920, 3735358488576, 28524555730944, 213934167982080, 1579821548175360, 11510128422420480, 82872924641427456, 590469588070170624, 4168020621671792640, 29176144351702548480

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (24,-216,864,-1296).

Cf. A000400, A007531, A036289, A081144, A128796.

Cf. A128960, A128961, A128962, A128963, A128965, A128967, A128969.

[(n^3-n)*6^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

I:=[0, 216, 5184, 77760]; [n le 4 select I[n] else 24*Self(n-1) -216*Self(n-2) +864*Self(n-3) -1296*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

CoefficientList[Series[216 x/(1 - 6 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 216*x^2/(1-6*x)^4.
a(n) = 216*A081144(n+1). (End)
a(n) = 24*a(n-1) - 216*a(n-2) + 864*a(n-3) - 1296*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=2} 1/a(n) = 25*log(6/5)/12 - 3/8.
Sum_{n>=2} (-1)^n/a(n) = 49*log(7/6)/12 - 5/8. (End)
a(n) = A007531(n+1)*A000400(n). - Amiram Eldar, Oct 02 2022

Corrected offset. - Mohammad K. Azarian, Nov 20 2008

A128985 a(n) = (n^3 - n^2)*2^n.

Original entry on oeis.org

0, 16, 144, 768, 3200, 11520, 37632, 114688, 331776, 921600, 2478080, 6488064, 16613376, 41746432, 103219200, 251658240, 606076928, 1443889152, 3406823424, 7969177600, 18496880640, 42630905856, 97626619904, 222264557568

Offset: 1

Mohammad K. Azarian, Apr 30 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A128796, A036289.

[(n^3-n^2)*2^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

I:=[0, 16, 144, 768]; [n le 4 select I[n] else 8*Self(n-1) - 24*Self(n-2) + 32*Self(n-3) - 16*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

CoefficientList[Series[16 x (1+x)/(1 - 2 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)
Table[(n^3-n^2)2^n,{n,30}] (* or *) LinearRecurrence[{8,-24,32,-16},{0,16,144,768},30] (* Harvey P. Dale, Jul 06 2014  *)

a(n)=(n^3-n^2)<Charles R Greathouse IV, Oct 07 2015

G.f.: 16*x^2*(1 + x)/(1 - 2*x)^4. - Vincenzo Librandi, Feb 11 2013

a(n) = 8*a(n-1) -24*a(n-2) +32*a(n-3) -16*a(n-4). - Vincenzo Librandi, Feb 11 2013

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A128961 a(n) = (n^3 - n)*3^n.

Original entry on oeis.org

0, 54, 648, 4860, 29160, 153090, 734832, 3306744, 14171760, 58458510, 233834040, 911952756, 3482001432, 13057505370, 48212327520, 175630621680, 632270238048, 2252462723046, 7949868434280, 27824539519980, 96653663595720, 333455139405234, 1143274763675088

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

Cf. A000244, A007531, A036289, A128796.

Cf. A128960, A128962, A128963, A128964, A128965, A128967, A128969.

[(n^3-n)*3^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

I:=[0,54,648,4860]; [n le 4 select I[n] else 12*Self(n-1)-54*Self(n-2)+108*Self(n-3)-81*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

LinearRecurrence[{12, -54, 108, -81}, {0, 54, 648, 4860}, 30] (* or *) CoefficientList[Series[54 x/(1 - 3 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)

G.f.: 54*x^2/(1-3*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 81*a(n-4). - Vincenzo Librandi, Feb 12 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000244(n).
Sum_{n>=2} 1/a(n) = (2/3)*log(3/2) - 1/4.
Sum_{n>=2} (-1)^n/a(n) = (8/3)*log(4/3) - 3/4. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128962 a(n) = (n^3 - n)*4^n.

Original entry on oeis.org

0, 96, 1536, 15360, 122880, 860160, 5505024, 33030144, 188743680, 1038090240, 5536481280, 28789702656, 146565758976, 732828794880, 3607772528640, 17523466567680, 84112639524864, 399535037743104, 1880164883496960, 8774102789652480, 40637949762600960

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Cf. A000302, A007531, A036289, A128796.

Cf. A128960, A128961, A128963, A128964, A128965, A128967, A128969.

[(n^3-n)*4^n: n in [1..20]]; // Vincenzo Librandi, Feb 09 2013

CoefficientList[Series[96 x / (1-4 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 09 2013 *)
Table[(n^3-n)4^n,{n,20}] (* or *) LinearRecurrence[{16,-96,256,-256},{0,96,1536,15360},20] (* Harvey P. Dale, Dec 31 2018 *)

G.f.: 96*x^2/(1-4*x)^4. - Vincenzo Librandi, Feb 09 2013
a(n) = 16*a(n-1) - 96*a(n-2) + 256*a(n-3) - 256*a(n-4). - Vincenzo Librandi, Feb 09 2013
a(n) = 96*A038846(n-2) for n>1. - Bruno Berselli, Feb 10 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000302(n).
Sum_{n>=2} 1/a(n) = (9/8)*log(4/3) - 5/16.
Sum_{n>=2} (-1)^n/a(n) = (25/8)*log(5/4) - 11/16. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128963 a(n) = (n^3 - n)*5^n.

Original entry on oeis.org

0, 150, 3000, 37500, 375000, 3281250, 26250000, 196875000, 1406250000, 9667968750, 64453125000, 418945312500, 2666015625000, 16662597656250, 102539062500000, 622558593750000, 3735351562500000, 22178649902343750, 130462646484375000, 761032104492187500

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A000351, A007531, A036289, A081143, A128796.

Cf. A128960, A128961, A128962, A128964, A128965, A128967, A128969.

[(n^3-n)*5^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

Table[(n^3-n)5^n,{n,20}] (* or *) LinearRecurrence[{20,-150,500,-625},{0,150,3000,37500},20] (* Harvey P. Dale, Jul 22 2012 *)
CoefficientList[Series[150 x/(1 - 5 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)

a(1)=0, a(2)=150, a(3)=3000, a(4)=37500, a(n)=20*a(n-1)-150*a(n-2)+ 500*a(n-3)- 625*a(n-4). - Harvey P. Dale, Jul 22 2012
G.f.: 150*x^2/(1 - 5*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 150*A081143(n+1). - Bruno Berselli, Feb 12 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000351(n).
Sum_{n>=2} 1/a(n) = (8/5)*log(5/4) - 7/20.
Sum_{n>=2} (-1)^n/a(n) = (18/5)*log(6/5) - 13/20. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128965 a(n) = (n^3 - n)*7^n.

Original entry on oeis.org

0, 294, 8232, 144060, 2016840, 24706290, 276710448, 2905459704, 29054597040, 279650496510, 2610071300760, 23751648836916, 211605598728888, 1851548988877770, 15951806673408480, 135590356723972080, 1138958996481365472, 9467596658251350486, 77968443067952298120

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (28,-294,1372,-2401).

Cf. A000420, A007531, A036289, A128796, A140107.

Cf. A128960, A128961, A128962, A128963, A128964, A128967, A128969.

[(n^3 - n)*7^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

LinearRecurrence[{28, -294, 1372, -2401}, {0, 294, 8232, 144060}, 30] (* Vincenzo Librandi, Feb 11 2013 *)
Table[(n^3-n)7^n,{n,20}] (* Harvey P. Dale, May 14 2020 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 294x^2/(1-7x)^4.
a(n) = 294*A140107(n-2). (End)
a(n) = 28*a(n-1) - 294*a(n-2) + 1372*a(n-3) - 2401*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000420(n).
Sum_{n>=2} 1/a(n) = (18/7)*log(7/6) - 11/28.
Sum_{n>=2} (-1)^n/a(n) = (32/7)*log(8/7) - 17/28. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A128967 a(n) = (n^3-n)*8^n.

Original entry on oeis.org

0, 384, 12288, 245760, 3932160, 55050240, 704643072, 8455716864, 96636764160, 1063004405760, 11338713661440, 117922622078976, 1200666697531392, 12006666975313920, 118219490218475520, 1148417904979476480, 11024811887802974208, 104735712934128254976

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (32,-384,2048,-4096).

Cf. A001018, A007531, A036289, A128796, A140802.

Cf. A128960, A128961, A128962, A128963, A128964, A128965, A128969.

[(n^3 - n)*8^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

LinearRecurrence[{32, -384, 2048, -4096}, {0, 384, 12288, 245760}, 30] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 384x^2/(1-8x)^4.
a(n) = 384*A140802(n-2). (End)
a(n) = 32*a(n-1) - 384*a(n-2) + 2048*a(n-3) - 4096*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A001018(n).
Sum_{n>=2} 1/a(n) = (49/16)*log(8/7) - 13/32.
Sum_{n>=2} (-1)^n/a(n) = (81/16)*log(9/8) - 19/32. (End)

Corrected the offset. - Mohammad K. Azarian, Nov 20 2008

A128969 a(n) = (n^3 - n)*9^n.

Original entry on oeis.org

0, 486, 17496, 393660, 7085880, 111602610, 1607077584, 21695547384, 278942752080, 3451916556990, 41422998683880, 484649084601396, 5551434969070536, 62453643402043530, 691794203838020640, 7560322370515511280, 81651481601567521824, 872650209616752889494

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (36,-486,2916,-6561).

Cf. A001019, A007531, A036289, A038291, A128796.

Cf. A128960, A128961, A128962, A128963, A128964, A128965, A128967.

[(n^3-n)*9^n: n in [0..25]]; // Vincenzo Librandi, Feb 11 2013

I:=[0, 486, 17496, 393660]; [n le 4 select I[n] else 36*Self(n-1) - 486*Self(n-2) + 2916*Self(n-3) - 6561*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

CoefficientList[Series[486 x/(1 - 9 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008 (Start)
G.f.: 486x^2/(1-9x)^4.
a(n) = 486*A038291(n+1,3). (End)
a(n) = 36*a(n-1) - 486*a(n-2) + 2916*a(n-3) - 6561*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A001019(n).
Sum_{n>=2} 1/a(n) = (32/9)*log(9/8) - 5/12.
Sum_{n>=2} (-1)^n/a(n) = (50/9)*log(10/9) - 7/12. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A097064 Expansion of (1 - 4x + 6x^2)/(1 - 2*x)^2.

Original entry on oeis.org

1, 0, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240, 22528, 49152, 106496, 229376, 491520, 1048576, 2228224, 4718592, 9961472, 20971520, 44040192, 92274688, 192937984, 402653184, 838860800, 1744830464, 3623878656, 7516192768, 15569256448, 32212254720, 66571993088

Offset: 0

Paul Barry, Jul 22 2004

Binomial transform of A097062.

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

Essentially the same as A036289.

Cf. A001787, A002162, A016578, A097062.

CoefficientList[Series[(1-4x+6x^2)/(1-2x)^2,{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{4,-4},{0,2},30]] (* Harvey P. Dale, May 26 2011 *)

a(n) = (n-1)*2^(n-1) + 3*0^n/2.
a(n) = 4*a(n-1) - 4*a(n-2), n>2.
a(n) = Sum_{k=0..n} binomial(n, k)*((2k-1)/2 + 3*(-1)^k/2).
a(n+1)/2 = A001787(n).
From Amiram Eldar, Oct 01 2022: (Start)
Sum_{n>=2} 1/a(n) = log(2) (A002162).
Sum_{n>=2} (-1)^n/a(n) = log(3/2) (A016578). (End)
E.g.f.: (3 - exp(2*x)*(1 - 2*x))/2. - Stefano Spezia, Feb 12 2023

A134401 Row sums of triangle A134400.

Original entry on oeis.org

1, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240, 22528, 49152, 106496, 229376, 491520, 1048576, 2228224, 4718592, 9961472, 20971520, 44040192, 92274688, 192937984, 402653184, 838860800, 1744830464, 3623878656, 7516192768

Offset: 0

Gary W. Adamson, Oct 23 2007

Essentially the same sequence as A036289.
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 187, 190, 250 and 442, lead to this sequence. For the central square these vectors lead to the companion sequence 2*A001792, for n >= 1 and a(0)=1. - Johannes W. Meijer, Aug 15 2010
Number of vertices on a partially truncated n-cube (column 1 of A271316). - Vincent J. Matsko, Apr 07 2016

			a(3) = 24 = sum of row 3 terms of triangle A134400: (3 + 9 + 9 + 3).
a(3) = 24 = (1, 3, 3, 1) dot (1, 1, 5, 5) = (1 + 3 + 15 + 5).

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A001792, A036289, A097064, A134400, A175654, A271316.

a:=Concatenation([1],List([1..30],n->n*2^n)); # Muniru A Asiru, Oct 28 2018

1,seq(n*2^n,n=1..30); # Muniru A Asiru, Oct 28 2018

F = Function[x, x*2^x];F[Range[1, 10]] (* Eugeny Yakimovitch (Eugeny.Yakimovitch(AT)gmail.com), Jan 08 2008 *)
{1}~Join~Table[n 2^n, {n, 28}] (* or *) Total /@ Join[{{1}}, Table[n Binomial[n, k], {n, 28}, {k, 0, n}]] (* Michael De Vlieger, Apr 07 2016 *)

x='x+O('x^99); Vec((1-2*x+4*x^2)/(1-2*x)^2) \\ Altug Alkan, Apr 07 2016

Binomial transform of repeats of (4n+1): [1, 1, 5, 5, 9, 9, 13, 13, ...].
a(n) = n*2^n, n > 1. - Eugeny Yakimovitch (Eugeny.Yakimovitch(AT)gmail.com), Jan 08 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) for n > 2.
G.f.: (1 - 2*x + 4*x^2)/(1-2*x)^2. (End)
E.g.f.: 1-E(0) where E(k)=1 - (k+1)/(1 - 2*x/(2*x - (k+1)^2/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2012
a(n) = A097064(n+1) for n >= 1. - Georg Fischer, Oct 28 2018
E.g.f.: 1 + 2*exp(2*x)*x. - Stefano Spezia, Feb 12 2023

More terms from Johannes W. Meijer, Aug 15 2010

cp's OEIS Frontend

A128964 a(n) = (n^3-n)*6^n.

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A128985 a(n) = (n^3 - n^2)*2^n.

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A128961 a(n) = (n^3 - n)*3^n.

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A128962 a(n) = (n^3 - n)*4^n.

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A128963 a(n) = (n^3 - n)*5^n.

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A128965 a(n) = (n^3 - n)*7^n.

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A128967 a(n) = (n^3-n)*8^n.

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A097064 Expansion of (1 - 4x + 6x^2)/(1 - 2*x)^2.