A008454 -id:A008454 - cp's OEIS frontend

A269792 a(n) = 5*n^4.

0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, 50000, 73205, 103680, 142805, 192080, 253125, 327680, 417605, 524880, 651605, 800000, 972405, 1171280, 1399205, 1658880, 1953125, 2284880, 2657205, 3073280, 3536405, 4050000, 4617605, 5242880, 5929605

Offset: 0

Ilya Gutkovskiy, Mar 31 2016

More generally, the ordinary generating function for the sequences of the form k*n^m, is k*Sum_{j>=1}x^j*j^m (when abs(x)<1).

More generally, the ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (p + q + k + m - 4*r)*x + (11*p + 3*q - k - 3*m + 6*r)*x^2 + (11*p - 3*q - k + 3*m - 4*r)*x^3 + (p - q + k - m + r)*x^4)/(1 - x)^5.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quartic polynomial
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. similar sequences of the form k*n^m, for k = 1...5, m = 1...10: A001477(k = 1, m = 1), A005843 (k = 2, m = 1), A008585 (k = 3, m = 1), A008586 (k = 4, m = 1), A008587 (k = 5, m = 1), A000290 (k = 1, m = 2), A001105 (k = 2, m = 2), A033428 (k = 3, m = 2), A016742 (k = 4, m = 2), A033429 (k = 5, m = 2), A000578 (k = 1, m = 3), A033431 (k = 2, m = 3), A117642 (k = 3, m = 3), A033430 (k = 4, m = 3), A244725 (k = 5, m = 3), A000583 (k = 1, m = 4), A244730 (k = 2, m = 4), A219056 (k = 3, m = 4), A141046 (k = 4, m = 4), this sequence(k = 5, m = 4), A000584 (k = 1, m = 5), A001014 (k = 1, m = 6), A106318 (k = 2, m = 6), A001015 (k = 1, m = 7), A001016 (k = 1, m = 8), A001017 (k = 1, m = 9), A008454 (k = 1, m = 10).

A269792:=n->5*n^4: seq(A269792(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[5 n^4, {n, 0, 33}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 80, 405, 1280}, 34]

x='x+O('x^99); concat(0, Vec(5*x*(1+11*x+11*x^2+x^3)/(1-x)^5)) \\ Altug Alkan, Mar 31 2016

G.f.: 5*x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5.
E.g.f.: 5*exp(x)^x*x*(1 + 7*x + 6*x^2 + x^3).
a(n) = 5*a(n-1) - 10*(9n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 5*A000583(n) = A008587(n)*A000578(n).
Sum_{n>=1} 1/a(n) = Pi^4/450 = (1/450)*A092425 = 0.216464646742...

A004897 Numbers that are the sum of at most 2 nonzero 10th powers.

Original entry on oeis.org

0, 1, 2, 1024, 1025, 2048, 59049, 59050, 60073, 118098, 1048576, 1048577, 1049600, 1107625, 2097152, 9765625, 9765626, 9766649, 9824674, 10814201, 19531250, 60466176, 60466177, 60467200, 60525225, 61514752, 70231801, 120932352

Offset: 1

N. J. A. Sloane

nonn

Zhuorui He, Table of n, a(n) for n = 1..10000

Cf. A008454 (10th powers), A004802 (sum of 2).

lista(nn) = setbinop((x,y)->x^10+y^10, [0..nn]); \\ Michel Marcus, Jul 02 2025

A016774 a(n) = (3*n)^10.

Original entry on oeis.org

0, 59049, 60466176, 3486784401, 61917364224, 576650390625, 3570467226624, 16679880978201, 63403380965376, 205891132094649, 590490000000000, 1531578985264449, 3656158440062976, 8140406085191601, 17080198121677824

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, 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. A008454 (n^10).

[(3*n)^10: n in [0..20]]; // Vincenzo Librandi, May 09 2011

(3*Range[0,30])^10 (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,59049,60466176,3486784401,61917364224,576650390625,3570467226624,16679880978201,63403380965376,205891132094649,590490000000000},30](* Harvey P. Dale, Mar 04 2013 *)

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11); a(0)=0, a(1)=59049, a(2)=60466176, a(3)=3486784401, a(4)=61917364224, a(5)=576650390625, a(6)=3570467226624, a(7)=16679880978201, a(8)=63403380965376, a(9)=205891132094649, a(10)=590490000000000. - Harvey P. Dale, Mar 04 2013

A016822 a(n) = (4n+1)^10.

Original entry on oeis.org

1, 9765625, 3486784401, 137858491849, 2015993900449, 16679880978201, 95367431640625, 420707233300201, 1531578985264449, 4808584372417849, 13422659310152401, 34050628916015625, 79792266297612001, 174887470365513049, 362033331456891249, 713342911662882601

Offset: 0

N. J. A. Sloane

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. A008454, A016813.

Table[(4n+1)^10,{n,0,100}] (* Mohammad K. Azarian, Jun 20 2016 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,9765625,3486784401,137858491849,2015993900449,16679880978201,95367431640625,420707233300201,1531578985264449,4808584372417849,13422659310152401},20] (* Harvey P. Dale, Sep 02 2025 *)

a(n) = A008454(A016813(n)). - Michel Marcus, Jun 21 2016

G.f.: (1 + 9765614*x + 3379362581*x^2 + 100040972648*x^3 + 689712304370*x^4 + 1514068354580*x^5 + 1167881384066*x^6 + 306865115624*x^7 + 22833444557*x^8 + 281825710*x^9 + 59049*x^10)/(1 - x)^11. - Ilya Gutkovskiy, Jun 21 2016

A016882 (5n+2)^10.

Original entry on oeis.org

1024, 282475249, 61917364224, 2015993900449, 26559922791424, 205891132094649, 1125899906842624, 4808584372417849, 17080198121677824, 52599132235830049, 144555105949057024, 362033331456891249, 839299365868340224, 1822837804551761449, 3743906242624487424

Offset: 0

N. J. A. Sloane

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. A008454, A016873.

A016882:=n->(5*n+2)^10; seq(A016882(n), n=0..30); # Wesley Ivan Hurt, Jan 22 2014

Table[(5n+2)^10, {n,0,30}] (* Wesley Ivan Hurt, Jan 22 2014 *)

a(n) = (5*n+2)^10; \\ Michel Marcus, Jan 27 2014

a(n) = A016873(n)^10 = A008454(A016873(n)). - Wesley Ivan Hurt, Jan 22 2014

A016918 a(n) = (6*n)^10.

Original entry on oeis.org

0, 60466176, 61917364224, 3570467226624, 63403380965376, 590490000000000, 3656158440062976, 17080198121677824, 64925062108545024, 210832519264920576, 604661760000000000, 1568336880910795776, 3743906242624487424

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A008454, A008588.

[(6*n)^10: n in [0..25]]; // Vincenzo Librandi, May 03 2011

(6*Range[0,20])^10 (* Harvey P. Dale, Jan 29 2013 *)

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). - Wesley Ivan Hurt, Jan 20 2024

A017170 a(n) = (9*n)^10.

Original entry on oeis.org

0, 3486784401, 3570467226624, 205891132094649, 3656158440062976, 34050628916015625, 210832519264920576, 984930291881790849, 3743906242624487424, 12157665459056928801, 34867844010000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

[(9*n)^10: n in [0..15]]; // Vincenzo Librandi, Jul 22 2011

a(n) = 3486784401*A008454(n). - R. J. Mathar, Jul 07 2017

A017362 a(n) = (10*n + 7)^10.

Original entry on oeis.org

282475249, 2015993900449, 205891132094649, 4808584372417849, 52599132235830049, 362033331456891249, 1822837804551761449, 7326680472586200649, 24842341419143568849, 73742412689492826049

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A017353 (10n+7), A008454 (n^10).

[(10*n+7)^10: n in [0..20]]; // Vincenzo Librandi, Aug 30 2011

(10*Range[0,10]+7)^10 (* Harvey P. Dale, Nov 12 2015 *)

vector(20, n, n--; (10*n+7)^10) \\ G. C. Greubel, Nov 10 2018

A017566 a(n) = (12*n+3)^10.

Original entry on oeis.org

59049, 576650390625, 205891132094649, 8140406085191601, 119042423827613001, 984930291881790849, 5631351470947265625, 24842341419143568849, 90438207500880449001, 283942098606901565601, 792594609605189126649

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A008454, A017557.

A017566:=n->(12*n+3)^10: seq(A017566(n), n=0..20); # Wesley Ivan Hurt, Apr 28 2017

(12Range[0,10]+3)^10 (* Harvey P. Dale, Aug 20 2012 *)

A017590 a(n) = (12*n+5)^10.

Original entry on oeis.org

9765625, 2015993900449, 420707233300201, 13422659310152401, 174887470365513049, 1346274334462890625, 7326680472586200649, 31181719929966183601, 110462212541120451001, 339456738992222314849

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A008454, A017581.

A017590:=(12*n+5)^10: seq(A017590(n), n=0..20); # Wesley Ivan Hurt, Apr 03 2017

(12*Range[0,20]+5)^10 (* Harvey P. Dale, Feb 19 2015 *)

cp's OEIS Frontend

A269792 a(n) = 5*n^4.

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A004897 Numbers that are the sum of at most 2 nonzero 10th powers.

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A016774 a(n) = (3*n)^10.

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A016822 a(n) = (4n+1)^10.

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A016882 (5n+2)^10.

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A016918 a(n) = (6*n)^10.

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Magma

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A017170 a(n) = (9*n)^10.

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A017362 a(n) = (10*n + 7)^10.

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A017566 a(n) = (12*n+3)^10.