A129002 -id:A129002 - cp's OEIS frontend

A128074 a(n) = (n^3+n)*9^n.

0, 18, 810, 21870, 446148, 7676370, 117979902, 1674039150, 22384294920, 285916320882, 3521652245010, 42113381995278, 491427393476940, 5617523480607090, 63094193590782438, 697970937800860110

Offset: 0

Mohammad K. Azarian, May 02 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (corrected by Ray Chandler, Jan 19 2019)
Index entries for linear recurrences with constant coefficients, signature (36,-486,2916,-6561).

Cf. A128796; A128960; A128985; A129002.

[(n^3 + n) * 9^n: n in [1..20]]; // Vincenzo Librandi, Feb 22 2012

Table[(n^3+n)9^n,{n,20}] (* or *) LinearRecurrence[{36,-486,2916,-6561}, {18,810,21870,446148},20] (* Harvey P. Dale, Jun 16 2011 *)

A128074(n)=(n^3+n)*9^n \\ M. F. Hasler, Oct 06 2014

a(1)=18, a(2)=810, a(3)=21870, a(4)=446148, a(n)=36*a(n-1)- 486*a(n-2)+ 2916*a(n-3)-6561*a(n-4). - Harvey P. Dale, Jun 16 2011

G.f.: 18*x*(1+9*x+81*x^2)/(1-9*x)^4. - Harvey P. Dale, Jun 16 2011

Extended to a(0)=0 by M. F. Hasler, Oct 06 2014

A352847 Number of copies of the star graph S(2,1,1) contained within the n-dimensional hypercube graph.

Original entry on oeis.org

0, 0, 48, 576, 3840, 19200, 80640, 301056, 1032192, 3317760, 10137600, 29736960, 84344832, 232587264, 626196480, 1651507200, 4278190080, 10909384704, 27433893888, 68136468480, 167352729600, 406931374080, 980510834688, 2343038877696, 5556613939200

Offset: 1

Ben Eck, Apr 05 2022

The star graph S(2,1,1) is the unique tree with degree sequence 3,2,1,1,1.

Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Cf. A129002.

a[n_] := (2^n)*Binomial[n,2]*(n-1)*(n-2); Array[a, 25] (* Amiram Eldar, Apr 22 2022 *)

from math import comb
def a(n):
    return (2**n)*comb(n,2)*(n-2)*(n-1)

a(n) = 2^n*binomial(n,2)*(n-1)*(n-2).
G.f.: 48*x^3*(1 + 2*x)/(1 - 2*x)^5. - Stefano Spezia, Apr 15 2022
Sum_{n>=3} 1/a(n) = 9/8 + log(2)^2/2 - 3*log(2)/4 - Pi^2/12. - Amiram Eldar, Apr 22 2022

A352994 Number of copies of the star graph S(2,2,1) contained within the n-dimensional hypercube graph.

Original entry on oeis.org

0, 0, 72, 1536, 14400, 92160, 470400, 2064384, 8128512, 29491200, 100362240, 324403200, 1005109248, 3005743104, 8722022400, 24662507520, 68183654400, 184817811456, 492285984768, 1291006771200, 3338686955520, 8526181171200, 21526669688832, 53788022931456

Offset: 1

Ben Eck, Apr 14 2022

S(2,2,1) is the star graph with two legs of length two and one of length one.

Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Cf. A129002, A352847.

a[n_] := (2^n)*Binomial[n,3]*3*n*(n-2); Array[a, 24] (* Amiram Eldar, Apr 22 2022 *)

from math import comb
def a(n):
    return (2**n)*comb(n,3)*3*n*(n-2)

a(n) = (2^n)*(C(n,3))*(3n)*(n-2).
G.f.: 24*x^3*(3 + 28*x + 12*x^2)/(1 - 2*x)^6. - Stefano Spezia, Apr 15 2022
Sum_{n>=3} 1/a(n) = 13/32 + 3*log(2)^2/16 - log(2)/4 - Pi^2/32. - Amiram Eldar, Apr 22 2022

A119635 a(n) = n(1 + n^2)2^n.

Original entry on oeis.org

4, 40, 240, 1088, 4160, 14208, 44800, 133120, 377856, 1034240, 2748416, 7127040, 18104320, 45187072, 111083520, 269484032, 646184960, 1533542400, 3606052864, 8409579520, 19465764864, 44753223680, 102257131520, 232330887168

Offset: 1

Mohammad K. Azarian, May 02 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, A128960, A128985, A129002.

List([1..30],n->n*(n^2+1)*2^n); # Muniru A Asiru, Mar 04 2019

[(n^3 + n)*2^n: n in [1..30]]; // Vincenzo Librandi, Feb 22 2013

[(n^3+n)*2^n$n=1..30]; # Muniru A Asiru, Mar 04 2019

Table[(n^3 + n)*2^n, {n, 30}] (* or *) CoefficientList[Series[4(1 +2x + 4x^2)/(1-2x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 22 2013 *)

{a(n) = n*(1+n^2)*2^n}; \\ G. C. Greubel, Mar 04 2019

[n*(1+n^2)*2^n for n in (1..30)] # G. C. Greubel, Mar 04 2019

G.f.: 4*x*(1 + 2*x + 4*x^2)/(1 - 2*x)^4. - Vincenzo Librandi, Feb 22 2013

E.g.f.: 4*x*(1 + 3*x + 2*x^2)*exp(2*x). - G. C. Greubel, Mar 04 2019

A121607 (n^3+n)*3^n.

Original entry on oeis.org

6, 90, 810, 5508, 31590, 161838, 765450, 3411720, 14526054, 59639490, 237731274, 924707340, 3523453830, 13191428502, 48642794730, 177008116752, 636661003590, 2266409860650, 7994034370026, 27964010896020, 97092998430246

Offset: 1

Mohammad K. Azarian, May 02 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A128796, A128960, A128985, A129002.

[(n^3 + n)*3^n: n in [1..30]]; // Vincenzo Librandi, Feb 22 2013

Table[(n^3 + n)*3^n, {n, 30}] (* or *) CoefficientList[Series[6 (1 + 3 x + 9 x^2)/(1 - 3 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 22 2013 *)

G.f.: 6*x*(1 + 3*x + 9*x^2)/(1 - 3*x)^4. - Vincenzo Librandi, Feb 22 2013

A127369 (n^3+n)*4^n.

Original entry on oeis.org

8, 160, 1920, 17408, 133120, 909312, 5734400, 34078720, 193462272, 1059061760, 5628755968, 29192355840, 148310589440, 740344987648, 3639984783360, 17660905521152, 84696755077120, 402008938905600, 1890610243960832

Offset: 1

Mohammad K. Azarian, May 02 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. A128796, A128960, A128985, A129002.

[(n^3 + n) * 4^n: n in [1..20]]; // Vincenzo Librandi, Feb 22 2013

Table[(n^3 + n) 4^n, {n, 20}] (* Vincenzo Librandi Feb 22 2013 *)
LinearRecurrence[{16,-96,256,-256},{8,160,1920,17408},20] (* Harvey P. Dale, Aug 14 2021 *)

G.f.: 8*x*(1+4*x+16*x^2)/(1-4*x)^4. [R. J. Mathar, Dec 19 2008]

A128013 a(n) = (n^3 +n)*5^n.

Original entry on oeis.org

10, 250, 3750, 42500, 406250, 3468750, 27343750, 203125000, 1441406250, 9863281250, 65527343750, 424804687500, 2697753906250, 16833496093750, 103454589843750, 627441406250000, 3761291503906250, 22315979003906250

Offset: 1

Mohammad K. Azarian, May 02 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. A128796, A128960, A128985, A129002.

[(n^3 + n)*5^n: n in [1..20]]; // Vincenzo Librandi, Feb 22 2013

I:=[10,250,3750,42500]; [n le 4 select I[n] else 20*Self(n-1)-150*Self(n-2)+500*Self(n-3)-625*Self(n-4): n in [1..20]]; // Vincenzo Librandi, Feb 23 2013

Table[(n^3 + n) 5^n, {n, 30}] (* or *) CoefficientList[Series[10 (1 + 5 x + 25 x^2)/(1 - 5 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 22 2013 *)

for(n=1, 30, print1((n^3 +n)*5^n, ", ")) \\ G. C. Greubel, May 08 2018

G.f.: 10*x(1+5*x+25*x^2)/(1-5*x)^4. - Vincenzo Librandi, Feb 22 2013

a(n) = 20*a(n-1) -150*a(n-2) +500*a(n-3) -625*a(n-4). - Vincenzo Librandi, Feb 23 2013

A128043 (n^3+n)*6^n.

Original entry on oeis.org

12, 360, 6480, 88128, 1010880, 10357632, 97977600, 873400320, 7437339648, 61070837760, 486873649152, 3787601264640, 28864133775360, 216128364576768, 1593927097712640, 11600403939459072, 83448431062548480

Offset: 1

Mohammad K. Azarian, May 02 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A128796, A128960, A128985, A129002.

[(n^3 + n)*6^n: n in [1..20]]; // Vincenzo Librandi, Feb 22 2013

Table[(n^3 + n) * 6^n, {n, 30}] (* or *) CoefficientList[Series[12 (1 + 6 x + 36 x^2)/(1 - 6 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 22 2013 *)

G.f.: 12*x*(1 + 6*x + 36*x^2)/(1 - 6*x)^4. - Vincenzo Librandi, Feb 22 2013

A128048 (n^3+n)*8^n.

Original entry on oeis.org

16, 640, 15360, 278528, 4259840, 58195968, 734003200, 8724152320, 99052683264, 1084479242240, 11527692222464, 119571889520640, 1214960348692480, 12129812277624832, 119275021381140480, 1157425104234217472

Offset: 1

Mohammad K. Azarian, May 02 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. A128796, A128960, A128985, A129002.

[(n^3 + n)*8^n: n in [1..20]]; // Vincenzo Librandi, Feb 22 2013

I:=[16,640,15360,278528]; [n le 4 select I[n] else 32*Self(n-1)-384*Self(n-2)+2048*Self(n-3)-4096*Self(n-4): n in [1..20]]; // Vincenzo Librandi, Feb 23 2013

Table[(n^3 + n) 8^n, {n, 30}] (* or *) CoefficientList[Series[16 (1 + 8 x + 64 x^2)/(1 - 8 x)^4, {x, 0, 20}],x] (* Vincenzo Librandi, Feb 22 2013 *)

G.f.: 16*x*(1+8*x+64*x^2)/(1-8*x)^4. - Vincenzo Librandi, Feb 22 2013

a(n) = 32*a(n-1) -384*a(n-2) +2048*a(n-3) -4096*a(n-4). - Vincenzo Librandi, Feb 23 2013

A128051 (n^3+n)*7^n.

Original entry on oeis.org

14, 490, 10290, 163268, 2184910, 26118078, 288240050, 2997696520, 29780961966, 285300001490, 2653572489106, 24083839729740, 214124712999470, 1870539234917542, 16094233518706770, 136653810502199312

Offset: 1

Mohammad K. Azarian, May 02 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A128796, A128960, A128985, A129002.

[(n^3 + n)*7^n: n in [1..20]]; // Vincenzo Librandi, Feb 22 2013

Table[(n^3 + n) * 7^n, {n, 30}] (* or *) CoefficientList[Series[14 (1 + 7 x + 49 x^2)/(1 - 7 x)^4, {x, 0, 20}], x] (* Vincenzo Librandi, Feb 22 2013 *)

G.f.: 14*x*(1 + 7*x + 49*x^2)/(1 - 7*x)^4. - Vincenzo Librandi, Feb 22 2013

cp's OEIS Frontend

A128074 a(n) = (n^3+n)*9^n.

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A352847 Number of copies of the star graph S(2,1,1) contained within the n-dimensional hypercube graph.

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A352994 Number of copies of the star graph S(2,2,1) contained within the n-dimensional hypercube graph.

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A119635 a(n) = n*(1 + n^2)*2^n.

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A121607 (n^3+n)*3^n.

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Magma

Mathematica

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A127369 (n^3+n)*4^n.

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Magma

Mathematica

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A128013 a(n) = (n^3 +n)*5^n.

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Magma

Magma

Mathematica

PARI

A119635 a(n) = n(1 + n^2)2^n.