A008454 -id:A008454 - cp's OEIS frontend

A089081 26th powers: a(n) = n^26.

0, 1, 67108864, 2541865828329, 4503599627370496, 1490116119384765625, 170581728179578208256, 9387480337647754305649, 302231454903657293676544, 6461081889226673298932241, 100000000000000000000000000

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Dec 04 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to divisibility sequences

Cf. A008454-A008456, A010801-A010813.

[n^26: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

Range[0, 9]^26 (* Alonso del Arte, Apr 26 2015 *)

a(n)=n^26 \\ Charles R Greathouse IV, Jun 28 2015

a(n) = n^26.
Completely multiplicative sequence with a(p) = p^26 for prime p. Multiplicative sequence with a(p^e) = p^(26e). - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 09 2020: (Start)
Dirichlet g.f.: zeta(s-26).
Sum_{n>=1} 1/a(n) = zeta(26) = 1315862*Pi^26/11094481976030578125.
Sum_{n>=1} (-1)^(n+1)/a(n) = 33554431*zeta(26)/33554432 = 22076500342261*Pi^26/186134520519971831808000000. (End)

A022526 Nexus numbers (n+1)^10-n^10.

Original entry on oeis.org

1, 1023, 58025, 989527, 8717049, 50700551, 222009073, 791266575, 2413042577, 6513215599, 15937424601, 35979939623, 75941127625, 151396163127, 287395735649, 522861237151, 916482272673, 1554473326175, 2560599031177, 4108933742199, 6439880978201, 9880041813223

Offset: 0

N. J. A. Sloane

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Column k=9 of A047969.
Cf. A008454 (n^10).
Cf. A008292, A022525, A022527.

[(n+1)^10-n^10: n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

b:=10: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018

Table[(n+1)^10-n^10,{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)

for(n=0,20, print1((n+1)^10 - n^10, ", ")) \\ G. C. Greubel, Feb 27 2018

G.f.: (x +1)*(x^8 +1012*x^7 +46828*x^6 +408364*x^5 +901990*x^4 +408364*x^3 +46828*x^2 +1012*x +1) / (x -1)^10. - Colin Barker, Dec 22 2012
a(n) = A008454(n+1) - A008454(n). - Michel Marcus, Feb 28 2018
G.f.: polylog(-10, x)*(1-x)/x. See the g.f. of Colin Barker (with expanded numerator), and the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

More terms from Colin Barker, Dec 22 2012

A016786 a(n) = (3*n+1)^10.

Original entry on oeis.org

1, 1048576, 282475249, 10000000000, 137858491849, 1099511627776, 6131066257801, 26559922791424, 95367431640625, 296196766695424, 819628286980801, 2064377754059776, 4808584372417849, 10485760000000000, 21611482313284249, 42420747482776576, 79792266297612001

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. A008454, A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784, A016785.

[(3*n+1)^10: n in [0..20]]; // Vincenzo Librandi, Sep 29 2011

Table[(3n+1)^10,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,1048576,282475249,10000000000,137858491849,1099511627776,6131066257801,26559922791424,95367431640625,296196766695424,819628286980801},20] (* Harvey P. Dale, May 14 2019 *)

a(n) = A008454(A016777(n)). - Michel Marcus, Jun 15 2016

Sum_{n>=0} 1/a(n) = PolyGamma(9, 1/3)/21427701120. - Amiram Eldar, Mar 29 2022

A016798 a(n) = (3*n + 2)^10.

Original entry on oeis.org

1024, 9765625, 1073741824, 25937424601, 289254654976, 2015993900449, 10240000000000, 41426511213649, 141167095653376, 420707233300201, 1125899906842624, 2758547353515625, 6278211847988224, 13422659310152401, 27197360938418176, 52599132235830049

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. A016789, A016790, A016791, A016792, A016793, A016794, A016795.

Subsequence of A008454.

(3*Range[0,20]+2)^10 (* Harvey P. Dale, Nov 28 2014 *)

From  Harvey P. Dale, Nov 28 2014: (Start)
G.f.: -(1/((x-1)^11))(x^10+1048565*x^9+270940968*x^8+6950443776*x^7+ 43221615834*x^6+86805830970*x^5+61387794480*x^4+14663204952*x^3+ 966376269*x^2+9754361*x+1024).
a(n) = 59049*n^10 + 393660*n^9 + 1180980*n^8 + 2099520*n^7 + 2449440*n^6 + 1959552*n^5 + 1088640*n^4 + 414720*n^3 + 103680*n^2 + 15360*n + 1024. [corrected by Amiram Eldar, Mar 31 2022] (End)
From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016789(n)^10 = A016790(n)^5 = A016793(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(9, 2/3)/21427701120. (End)
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, Dec 31 2023

A240933 a(n) = n^10 - n^9.

Original entry on oeis.org

0, 0, 512, 39366, 786432, 7812500, 50388480, 242121642, 939524096, 3099363912, 9000000000, 23579476910, 56757583872, 127253992476, 268593608192, 538207031250, 1030792151040, 1897406023952, 3372107936256, 5808378560022, 9728000000000, 15885600931620, 25352653573632

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 10-digit positive integers in base n.

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

Cf. A001017, A008454.

Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240931, A240932.

[n^10-n^9 : n in [0..30]]; // Wesley Ivan Hurt, Aug 03 2014

A240933:=n->n^10-n^9: seq(A240933(n), n=0..30); # Wesley Ivan Hurt, Aug 03 2014

Table[n^10 - n^9, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 03 2014 *)
CoefficientList[Series[2 (256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,0,512,39366,786432,7812500,50388480,242121642,939524096,3099363912,9000000000},40] (* Harvey P. Dale, Oct 19 2022 *)

vector(100, n, (n-1)^10 - (n-1)^9) \\ Derek Orr, Aug 03 2014

a(n) = n^9*(n-1) = n^10 - n^9.
a(n) = A008454(n) - A001017(n). - Michel Marcus, Aug 03 2014
G.f.: 2*(256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11. - Wesley Ivan Hurt, Aug 03 2014
Recurrence: 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, Aug 03 2014
Sum_{n>=2} 1/a(n) = 9 - Sum_{k=2..9} zeta(k). - Amiram Eldar, Jul 05 2020

A016954 a(n) = (6n+3)^10.

Original entry on oeis.org

59049, 3486784401, 576650390625, 16679880978201, 205891132094649, 1531578985264449, 8140406085191601, 34050628916015625, 119042423827613001, 362033331456891249, 984930291881790849, 2446194060654759801, 5631351470947265625, 12157665459056928801

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, A016762, A016945, A016946, A016947, A016948, A016949, A016950, A016951, A016952, A016953.

Subsequence of A008454.

[(6*n+3)^10: n in [0..20]]; // Vincenzo Librandi, May 06 2011

A016954:=n->(6*n+3)^10: seq(A016954(n), n=0..20); # Wesley Ivan Hurt, Aug 22 2016

(6 Range[0, 20] + 3)^10 (* Wesley Ivan Hurt, Aug 22 2016 *)

From Wesley Ivan Hurt, Aug 22 2016: (Start)
G.f.: 59049*(1 + 59038*x + 9116141*x^2 + 178300904*x^3 + 906923282*x^4 + 1527092468*x^5 + 906923282*x^6 + 178300904*x^7 + 9116141*x^8 + 59038*x^9 + x^10)/(1-x)^11.
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) for n>10.
a(n) = A008454(A016945(n)). (End)
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016946(n)^5 = A016949(n)^2.
a(n) = 3^10*A016762(n).
Sum_{n>=0} 1/a(n) = 31*Pi^10/171421608960. (End)

A016966 a(n) = (6*n + 4)^10.

Original entry on oeis.org

1048576, 10000000000, 1099511627776, 26559922791424, 296196766695424, 2064377754059776, 10485760000000000, 42420747482776576, 144555105949057024, 430804206899405824, 1152921504606846976, 2824752490000000000, 6428888932339941376, 13744803133596058624

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. A016798, A016957, A016958, A016959, A016960, A016961, A016962, A016963, A016964, A016965.

Subsequence of A008454.

[(6*n+4)^10: n in [0..20]]; // Vincenzo Librandi, May 07 2011

(6Range[0,20]+4)^10  (* Harvey P. Dale, Feb 23 2011 *)

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016957(n)^10 = A016958(n)^5 = A016961(n)^2.
a(n) = 2^10*A016798(n).
Sum_{n>=0} 1/a(n) = PolyGamma(9, 2/3)/21941965946880. (End)

A016978 a(n) = (6*n + 5)^10.

Original entry on oeis.org

9765625, 25937424601, 2015993900449, 41426511213649, 420707233300201, 2758547353515625, 13422659310152401, 52599132235830049, 174887470365513049, 511116753300641401, 1346274334462890625, 3255243551009881201, 7326680472586200649, 15516041187205853449

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. A016969 (6*n + 5), A016970, A016971, A016972, A016973, A016974, A016975, A016976, A016977.

Subsequence of A008454 (n^10).

[(6*n+5)^10: n in [0..30]]; // Vincenzo Librandi, May 14 2011

(6*Range[0,20]+5)^10 (* Harvey P. Dale, Aug 26 2012 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^10 = A016970(n)^5 = A016973(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(9, 5/6)/21941965946880. (End)

A071236 a(n) = (n^10 + n^5)/2.

Original entry on oeis.org

0, 1, 528, 29646, 524800, 4884375, 30236976, 141246028, 536887296, 1743421725, 5000050000, 12968792826, 30958806528, 68929431571, 144627596400, 288325575000, 549756338176, 1007997660153, 1785234558096, 3065534366950, 5120001600000, 8339942531151

Offset: 0

N. J. A. Sloane, Jun 12 2002

Subset of A000217. - Robert Israel, Nov 20 2014

Number of unoriented rows of length 10 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=528, there are 2^10=1024 oriented arrangements of two colors. Of these, 2^5=32 are achiral. That leaves (1024-32)/2=496 chiral pairs. Adding achiral and chiral, we get 528. - Robert A. Russell, Nov 13 2018

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

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. A000217.
Row 10 of A277504.
Cf. A008454 (oriented), A000584 (achiral).

List([0..40], n -> n^5*(1 + n^5)/2); # G. C. Greubel, Nov 15 2018

[n^5*(n+1)*(n^4-n^3+n^2-n+1)/2: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

seq((n^10 + n^5)/2, n=0..100); # Robert Israel, Nov 19 2014

Table[n^5(n+1)(n^4-n^3+n^2-n+1)/2,{n,0,30}] (* or *) LinearRecurrence[{11, -55, 165, -330,462,-462,330,-165,55,-11,1},{0, 1, 528,29646,524800,4884375, 30236976,141246028,536887296, 1743421725, 5000050000}, 30](* Harvey P. Dale, Jul 24 2012 *)

a(n)=binomial(n^5+1,2) \\ Charles R Greathouse IV, Nov 19 2014

[n^5*(1 + n^5)/2 for n in range(40)] # G. C. Greubel, Nov 15 2018

a(0)=0, a(1)=1, a(2)=528, a(3)=29646, a(4)=524800, a(5)=4884375, a(6)=30236976, a(7)=141246028, a(8)=536887296, a(9)=1743421725, a(10)=5000050000, 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). - Harvey P. Dale, Jul 24 2012
From Robert Israel, Nov 19 2014: (Start)
a(n) = (A008454(n) + A000584(n))/2 = n^5*(n+1)*(n^4 -n^3 +n^2 -n +1)/2.
G.f.: x*(1 +517*x +23893*x^2 +227569*x^3 +655315*x^4 +655039*x^5 +227623*x^6 +23947*x^7 +496*x^8)/(1-x)^11. (End)
From Robert A. Russell, Nov 13 2018: (Start)
G.f.: (Sum_{j=1..10} S2(10,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..5} S2(5,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..9} A145882(10,k) * x^k / (1-x)^11.
E.g.f.: (Sum_{k=1..10} S2(10,k)*x^k + Sum_{k=1..5} S2(5,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n > 10, a(n) = Sum_{j=1..11} -binomial(j-12,j) * a(n-j). (End)
E.g.f.: x*(2 + 526*x + 9355*x^2 + 34115 x^3 + 42526*x^4 + 22827*x^5 + 5880*x^6 + 750*x^7 + 45*x^8 + x^9)*exp(x)/2. - G. C. Greubel, Nov 15 2018

Definition simplified by Robert Israel, Nov 19 2014

A198479 a(n) = 10^n * n^10.

Original entry on oeis.org

0, 10, 102400, 59049000, 10485760000, 976562500000, 60466176000000, 2824752490000000, 107374182400000000, 3486784401000000000, 100000000000000000000, 2593742460100000000000, 61917364224000000000000

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A008454, A011557.

Cf. A007758, A062074, A062075, A097206, A098803, A098880.

Magma
```
[10^n*n^10: n in [0..20]]
```

Table[10^n*n^10, {n,0,20}] (* G. C. Greubel, May 17 2022 *)

[10^n*n^10 for n in (0..20)] # G. C. Greubel, May 17 2022

G.f.: 10*x*(1 + 10*x)*(1 + 10120*x + 4682800*x^2 + 408364000*x^3 + 9019900000*x^4 + 40836400000*x^5 + 46828000000*x^6 + 10120000000*x^7 + 100000000*x^8)/ (1-10*x)^11. - Colin Barker, May 01 2013

a(n) = A011557(n)*A008454(n). - Michel Marcus, May 18 2022

cp's OEIS Frontend

A089081 26th powers: a(n) = n^26.

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A022526 Nexus numbers (n+1)^10-n^10.

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A016786 a(n) = (3*n+1)^10.

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A016798 a(n) = (3*n + 2)^10.

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A240933 a(n) = n^10 - n^9.

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A016954 a(n) = (6n+3)^10.

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A016966 a(n) = (6*n + 4)^10.

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A016978 a(n) = (6*n + 5)^10.