A001017 -id:A001017 - cp's OEIS frontend

A002239 9th powers written backwards.

0, 1, 215, 38691, 441262, 5213591, 69677001, 70635304, 827712431, 984024783, 1, 1967497532, 2530879515, 37399440601, 48764016602, 57395334483, 63767491786, 794678785811, 863092953891, 977796786223, 215, 185640082497, 2977129627021, 3641662511081, 4220457081462

Offset: 0

N. J. A. Sloane

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001017.

a:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||(n^9)):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 09 2015

A016797 a(n) = (3*n + 2)^9.

Original entry on oeis.org

512, 1953125, 134217728, 2357947691, 20661046784, 118587876497, 512000000000, 1801152661463, 5429503678976, 14507145975869, 35184372088832, 78815638671875, 165216101262848, 327381934393961, 618121839509504, 1119130473102767, 1953125000000000, 3299763591802133

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016789, A016790, A016791, A016792, A016793, A016794, A016795, A016796.

Subsequence of A001017.

Table[(3n+2)^9,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)

G.f.: (512 + 1948005*x + 114709518*x^2 + 1103599596*x^3 + 2887100154*x^4 + 2388954618*x^5 + 608260290*x^6 + 37732212*x^7 + 262134*x^8 + x^9)/(1 - x)^10. - Ilya Gutkovskiy, Jun 16 2016
From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016789(n)^9.
Sum_{n>=0} 1/a(n) = 9841*zeta(9)/19683 - 1618*Pi^9/(55801305*sqrt(3)). (End)

A168372 a(n) = n^5*(n^4 + 1)/2.

Original entry on oeis.org

0, 1, 272, 9963, 131584, 978125, 5042736, 20185207, 67125248, 193739769, 500050000, 1179054371, 2580014592, 5302435333, 10330792304, 19222059375, 34360262656, 59294648177, 99180589968, 161345086939, 256001600000

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 9 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)=272, there are 2^9=512 oriented arrangements of two colors. Of these, 2^5=32 are achiral. That leaves (512-32)/2=240 chiral pairs. Adding achiral and chiral, we get 272. - Robert A. Russell, Nov 13 2018

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).

Row 9 of A277504.

Cf. A001017 (oriented), A000584 (achiral).

List([0..30], n -> n^5*(n^4 + 1)/2); # G. C. Greubel, Nov 15 2018

[n^5*(n^4+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 28 2011

Table[n^5*(n^4 + 1)/2, {n,0,50}] (* G. C. Greubel, Jul 19 2016 *)

a(n)=n^5*(n^4+1)/2 \\ Charles R Greathouse IV, Jul 19 2016

[n^5*(1 + n^4)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: x*(1 + 262*x + 7288*x^2 + 44074*x^3 + 78190*x^4 + 44074*x^5 + 7288*x^6 + 262*x^7 + x^8)/(1 - x)^10. - G. C. Greubel, Jul 19 2016
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A001017(n) + A000584(n)) / 2 = (n^9 + n^5) / 2.
G.f.: (Sum_{j=1..9} S2(9,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..8} A145882(9,k) * x^k / (1-x)^10.
E.g.f.: (Sum_{k=1..9} S2(9,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>9, a(n) = Sum_{j=1..10} -binomial(j-11,j) * a(n-j). (End)
E.g.f.: x*(2 +270*x +3050*x^2 +7780*x^3 +6952*x^4 +2646*x^5 +462*x^6 + 36*x^7 +x^8)*exp(x)/2. - G. C. Greubel, Nov 15 2018

A198478 a(n) = 9^n * n^9.

Original entry on oeis.org

0, 9, 41472, 14348907, 1719926784, 115330078125, 5355700839936, 193010051319183, 5777633090469888, 150094635296999121, 3486784401000000000, 73994897046174912819, 1457274373159131021312, 26955214582765006137717

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001017, A001019.

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

Magma
```
[9^n*n^9: n in [0..20]]
```

Table[9^n*n^9, {n, 0, 20}] (* G. C. Greubel, May 17 2022 *)

[9^n*n^9 for n in (0..20)] # G. C. Greubel, May 17 2022

G.f.: 9*x*(1 + 4518*x + 1183248*x^2 + 64322586*x^3 + 1024762590*x^4 + 5210129466*x^5 + 7763290128*x^6 + 2401050438*x^7 + 43046721*x^8)/(1 - 9*x)^10. - Colin Barker, Apr 30 2013

a(n) = A001019(n)*A001017(n). - Michel Marcus, May 18 2022

A017169 a(n) = (9*n)^9.

Original entry on oeis.org

0, 387420489, 198359290368, 7625597484987, 101559956668416, 756680642578125, 3904305912313344, 15633814156853823, 51998697814228992, 150094635296999121, 387420489000000000, 913517247483640899

Offset: 0

N. J. A. Sloane

a(n) = A155955(n,9) for n > 8. - Reinhard Zumkeller, Jan 31 2009

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).

[(9*n)^9: n in [0..20]]; // Vincenzo Librandi, Jul 22 2011

(9*Range[0,20])^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,387420489,198359290368,7625597484987,101559956668416,756680642578125,3904305912313344,15633814156853823,51998697814228992,150094635296999121},20] (* Harvey P. Dale, Dec 11 2017 *)

a(n) = 387420489*A001017(n). - R. J. Mathar, Jul 07 2017

A075670 Sum of next n 9th powers.

Original entry on oeis.org

1, 20195, 12292965, 1561991824, 77226633575, 2014634387961, 33098483802475, 383318212734080, 3377498614484589, 23898971839102975, 141290020118952881, 719054471032657200, 3223613105991831475, 12964037775857022869, 47453810583528962775, 159982264435790734336

Offset: 1

Zak Seidov, Sep 24 2002

			a(1) = 1^9 = 1; a(2) = 2^9 + 3^9 = 20195; a(3) = 4^9 + 5^9 + 6^9 = 12292965; a(4) = 7^9 + 8^9 + 9^9 + 10^9 = 1561991824.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).

Cf. A001017 (9th powers).

Cf. A006003, A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).

[(5*n^19 + 105*n^17 + 666*n^15 + 1530*n^13 + 689*n^11 - 995*n^9 + 304*n^7 + 640*n^5 - 384*n^3)/2560 : n in [1..20]]; // Vincenzo Librandi, Oct 06 2011

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=9; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
Total[#^9]&/@(Range[First[#]+1,Last[#]]&/@Partition[Accumulate[Range[ 0,15]],2,1]) (* Harvey P. Dale, Oct 05 2011 *)
With[{nn=20},Total/@TakeList[Range[(nn(nn+1))/2]^9,Range[nn]]] (* Harvey P. Dale, Aug 05 2025 *)

a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^9.
a(n) = (5n^19 + 105n^17 + 666n^15 + 1530n^13 + 689n^11 - 995n^9 + 304n^7 + 640n^5 - 384n^3)/2560. - Charles R Greathouse IV, Sep 17 2009
G.f.: x*(x^18 +20175*x^17 +11889255*x^16 +1319968434*x^15 +48299442990*x^14 +752964012192*x^13 +5757432094050*x^12 +23468751060270*x^11 +53583908362248*x^10 +70362713036770*x^9 +53583908362248*x^8 +23468751060270*x^7 +5757432094050*x^6+752964012192*x^5 +48299442990*x^4 +1319968434*x^3 +11889255*x^2 +20175*x +1)/(x -1)^20. - Colin Barker, Sep 06 2012

A134010 a(n) = n^(initial digit of n).

Original entry on oeis.org

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 2560000, 2825761, 3111696

Offset: 0

Reinhard Zumkeller, Oct 02 2007

a(n) = n^A000030(n).

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A001017, A001016, A001015, A001014, A000584, A000583, A000578, A000290, A000027.

a[n_]:= n^First[IntegerDigits[n]];Join[{1},Array[a,42]] (* James C. McMahon, Mar 30 2025 *)

A170790 a(n) = n^9*(n^8 + 1)/2.

Original entry on oeis.org

0, 1, 65792, 64579923, 8590065664, 381470703125, 8463334761216, 116315277170407, 1125899973951488, 8338591043543529, 50000000500000000, 252723515428620731, 1109305555950108672, 4325207964992918653

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 17 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)=65792, there are 2^17=131072 oriented arrangements of two colors. Of these, 2^9=512 are achiral. That leaves (131072-512)/2=65280 chiral pairs. Adding achiral and chiral, we get 65792. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (18,-153,816,-3060,8568,-18564, 31824,-43758,48620,-43758,31824,-18564,8568,-3060,816,-153,18,-1).

Row 17 of A277504.

Cf. A010805 (oriented), A001017 (achiral).

List([0..30], n -> n^9*(n^8+1)/2); # G. C. Greubel, Nov 15 2018

[n^9*(n^8+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

Table[(n^9 (n^8+1))/2,{n,0,20}] (* Harvey P. Dale, Oct 03 2016 *)

for(n=0,30, print1(n^9*(n^8+1)/2, ", ")) \\ G. C. Greubel, Dec 06 2017

for n in range(0,20): print(int(n**9*(n**8 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^9*(n^8+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 65774*x^2 + 63395820*x^3 + 7437692410*x^4 + 236676566180*x^5 + 2858646249342*x^6 + 15527826341908*x^7 + 41568611082650*x^8 + 57445191259830*x^9 + 41568611082650*x^10 + 15527826341908*x^11 + 2858646249342*x^12 + 236676566180*x^13 + 7437692410*x^14 + 63395820*x^15 + 65774*x^16 + x^17)/(1-x)^18. - G. C. Greubel, Dec 06 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010805(n) + A001017(n)) / 2 = (n^17 + n^9) / 2.
G.f.: (Sum_{j=1..17} S2(17,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..9} S2(9,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..16} A145882(17,k) * x^k / (1-x)^18.
E.g.f.: (Sum_{k=1..17} S2(17,k)*x^k + Sum_{k=1..9} S2(9,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>17, a(n) = Sum_{j=1..18} -binomial(j-19,j) * a(n-j). (End)

A170791 a(n) = n^9*(n^9 + 1)/2.

Original entry on oeis.org

0, 1, 131328, 193720086, 34359869440, 1907349609375, 50779983373056, 814206819132028, 9007199321849856, 75047317842209805, 500000000500000000, 2779958657925089586, 13311666643022512128, 56227703481280946251

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 18 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)=131328, there are 2^18=262144 oriented arrangements of two colors. Of these, 2^9=512 are achiral. That leaves (262144-512)/2=130816 chiral pairs. Adding achiral and chiral, we get 131328. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (19,-171,969,-3876,11628,-27132, 50388,-75582,92378,-92378,75582,-50388,27132,-11628,3876,-969,171, -19,1).

Row 18 of A277504.

Cf. A010806 (oriented), A001017 (achiral).

List([0..30], n -> n^9*(n^9 + 1)/2); # G. C. Greubel, Nov 15 2018

[n^9*(n^9+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

f[n_]:=Module[{n9=n^9},(n9(n9+1))/2]; Array[f,20,0] (* Harvey P. Dale, Nov 24 2012 *)
Table[n^9*(n^9+1)/2, {n,0,30}] (* G. C. Greubel, Dec 06 2017 *)

for(n=0,30, print1(n^9*(n^9+1)/2, ", ")) \\ G. C. Greubel, Dec 06 2017

for n in range(0,20): print(int(n**9*(n**9 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^9*(1 + n^9)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 131309*x^2 + 191225025*x^3 + 30701643925*x^4 + 1287510971765*x^5 + 20228672721537*x^6 + 142998536758213*x^7 + 503354983579865*x^8 + 932692830330915*x^9 + 932692827449735*x^10 + 503354984335363*x^11 + 142998537549087*x^12 + 20228672026535*x^13 + 1287511125835*x^14 + 30701669175*x^15 + 191214899*x^16 + 130816*x^17) /(1-x)^19. - G. C. Greubel, Dec 06 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010806(n) + A001017(n)) / 2 = (n^18 + n^9) / 2.
G.f.: (Sum_{j=1..18} S2(18,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..9} S2(9,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..17} A145882(18,k) * x^k / (1-x)^19.
E.g.f.: (Sum_{k=1..18} S2(18,k)*x^k + Sum_{k=1..9} S2(9,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>18, a(n) = Sum_{j=1..19} -binomial(j-20,j) * a(n-j). (End)

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

Original entry on oeis.org

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...

cp's OEIS Frontend

A002239 9th powers written backwards.

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Maple

A016797 a(n) = (3*n + 2)^9.

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A168372 a(n) = n^5*(n^4 + 1)/2.

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A198478 a(n) = 9^n * n^9.

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A017169 a(n) = (9*n)^9.

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Magma

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A075670 Sum of next n 9th powers.

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Magma

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A134010 a(n) = n^(initial digit of n).

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Mathematica

A170790 a(n) = n^9*(n^8 + 1)/2.

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