A053699 -id:A053699 - cp's OEIS frontend

A259264 Cyclotomic polynomial value Phi(5,n!).

5, 5, 31, 1555, 346201, 209102521, 269112327121, 645369332030641, 2642973843312270721, 17340169097630273489281, 173401260912466521665068801, 2538767225004530212838194156801, 52643875968757162157863810977561601

Offset: 0

Robert Price, Jun 22 2015

nonn

Robert Price, Table of n, a(n) for n = 0..49
OEIS Wiki, Cyclotomic Polynomials at x=sigma(n)

Cf. A053699, A200906.

Mathematica
```
Table[Cyclotomic[5, n!], {n, 0, 200}]
```

vector(15, n, n--; polcyclo(5, n!)) \\ Michel Marcus, Jun 23 2015

a(n) = A053699(n!). - Michel Marcus, Jun 23 2015

A102909 a(n) = Sum_{j=0..8} n^j.

Original entry on oeis.org

1, 9, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961, 48427561, 111111111, 235794769, 469070941, 883708281, 1589311291, 2745954241, 4581298449, 7411742281, 11668193551, 17927094321, 26947368421, 39714002329, 57489010371, 81870575521, 114861197401

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Mar 01 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A001016, A060890, A059839.

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), this sequence (m=8), A103623 (m=9), A060885 (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..8]]): n in [0..30]]; // G. C. Greubel, Feb 13 2018

1 + Sum[Range[0, 30]^j, {j, 1, 8}] (* G. C. Greubel, Feb 13 2018 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,9,511,9841,87381,488281,2015539,6725601,19173961},30] (* Harvey P. Dale, Feb 01 2025 *)

a(n)=n^8+n^7+n^6+n^5+n^4+n^3+n^2+n+1 \\ Charles R Greathouse IV, Oct 07 2015

[sum(n^j for j in (0..8)) for n in (0..30)] # G. C. Greubel, Apr 14 2019

a(n) = (n^2+n+1) * (n^6+n^3+1) and so is never prime. - Jonathan Vos Post, Dec 21 2012
G.f.: (x^8 + 162*x^7 + 3418*x^6 + 14212*x^5 + 16578*x^4 + 5482*x^3 + 466*x^2 + 1)/(1-x)^9. - Colin Barker, Nov 05 2012, edited by M. F. Hasler, Dec 31 2012
a(n) = (n^9-1)/(n-1) with a(1) = 9. - L. Edson Jeffery and M. F. Hasler, Dec 30 2012
E.g.f.: exp(x)*(1 + 8*x + 247*x^2 + 1389*x^3 + 2127*x^4 + 1206*x^5 + 288*x^6 + 29*x^7 + x^8). - Stefano Spezia, Oct 03 2024

Offset corrected by N. J. A. Sloane, Dec 30 2012

A104376 a(n) = Sum_{j=0..13} n^j.

Original entry on oeis.org

1, 14, 16383, 2391484, 89478485, 1525878906, 15672832819, 113037178808, 628292358729, 2859599056870, 11111111111111, 37974983358324, 116719860413533, 328114698808274, 854769755812155, 2085209001813616, 4803839602528529, 10523614159962558, 22047845151887119

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Apr 16 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), A060885 (m=10), A105067 (m=11), A060887 (m=12), this sequence (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..13]]): n in [0..20]]; // Vincenzo Librandi, May 01 2011

Table[1+Sum[n^j, {j,1,13}], {n,0,20}] (* G. C. Greubel, Apr 14 2019 *)
LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,14,16383,2391484,89478485,1525878906,15672832819,113037178808,628292358729,2859599056870,11111111111111,37974983358324,116719860413533,328114698808274},20] (* Harvey P. Dale, Sep 04 2023 *)

a(n)=sum(j=0,13, n^j) \\ Charles R Greathouse IV, Oct 07 2015

[sum(n^j for j in (0..13)) for n in (0..20)] # G. C. Greubel, Apr 14 2019

a(n) = n^13 + n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n^1 + 1.

G.f.: (5461*x^12 + 1119288*x^11 + 37443654*x^10 + 372458048*x^9 + 1409085783*x^8 + 2263446576*x^7 + 1598944452*x^6 + 484853760*x^5 + 57484467*x^4 + 2163032*x^3 + 16278*x^2 + 1)/(1-x)^14. - Colin Barker, Nov 04 2012

Name changed by G. C. Greubel, Apr 14 2019

A104682 a(n) = Sum_{j=0..14} n^j.

Original entry on oeis.org

1, 15, 32767, 7174453, 357913941, 7629394531, 94036996915, 791260251657, 5026338869833, 25736391511831, 111111111111111, 417724816941565, 1400638324962397, 4265491084507563, 11966776581370171, 31278135027204241, 76861433640456465, 178901440719363487

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Apr 22 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), A060885 (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), this sequence (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..14]]): n in [0..20]]; // Vincenzo Librandi, May 01 2011

With[{f=Total[n^Range[0,14]]},Table[f,{n,0,20}]] (* Harvey P. Dale, Jun 11 2011 *)

a(n) = sum(j=0, 14, n^j) \\ Charles R Greathouse IV, Oct 07 2015

[sum(n^j for j in (0..14)) for n in (0..20)] # G. C. Greubel, Apr 15 2019

a(n) = n^14 + n^13 + n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n^1 + 1.
a(n) = (n^2 + n + 1) * (n^4 + n^3 + n^2 + n + 1) * (n^8 - n^7 + n^5 - n^4 + n^3 - n + 1). - Jonathan Vos Post, Apr 23 2005
G.f.: (x^14 +10908*x^13 +3423487*x^12 +162086420*x^11 +2236727781*x^10 +11806635128*x^9 +27116815299*x^8 +28635678216*x^7 +13957353555*x^6 +2999111468*x^5 +253732221*x^4 +6684068*x^3 +32647*x^2 +1)/(1-x)^15. - Colin Barker, Nov 04 2012

More terms from Harvey P. Dale, Jun 11 2011

Name changed by G. C. Greubel, Apr 15 2019

A105067 a(n) = Sum_{j=0..11} n^j.

Original entry on oeis.org

1, 12, 4095, 265720, 5592405, 61035156, 435356467, 2306881200, 9817068105, 35303692060, 111111111111, 313842837672, 810554586205, 1941507093540, 4361070182715, 9267595563616, 18764998447377, 36413889826860

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Apr 05 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), A060885 (m=10), this sequence (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1: n in [0..20]]; // Vincenzo Librandi, May 01 2011

1+Sum[Range[0,20]^j, {j,1,11}] (* G. C. Greubel, Apr 13 2019 *)

a(n)=polcyclo(11,n)+n^11 \\ Charles R Greathouse IV, Sep 03 2011

[sum(n^j for j in (0..11)) for n in (0..20)] # G. C. Greubel, Apr 13 2019

Factorization of the polynomial into irreducible components over integers: n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 = +- (n + 1) * (n^2 - n + 1) * (n^2 + 1) * (n^2 + n + 1) * (n^4 - n^2 + 1). - Jonathan Vos Post, Apr 06 2005

G.f.: (1365*x^10 + 116480*x^9 + 1851213*x^8 + 8893248*x^7 + 15593370*x^6 + 10568064*x^5 + 2671890*x^4 + 217152*x^3 + 4017*x^2 + 1)/(x - 1)^12. - Colin Barker, Oct 29 2012

Signature changed by Georg Fischer, Apr 13 2019

A190527 Primes of the form p^4 + p^3 + p^2 + p + 1, where p is prime.

Original entry on oeis.org

31, 2801, 30941, 88741, 292561, 732541, 3500201, 28792661, 39449441, 48037081, 262209281, 1394714501, 2666986681, 3276517921, 4802611441, 5908670381, 12936304421, 16656709681, 19408913261, 24903325661, 37226181521, 43713558101, 52753304641, 64141071121, 96427561501, 100648118041

Offset: 1

Bernard Schott, Dec 20 2012

nonn

These primes are generated by exactly A065509, cf. 2nd formula.
These numbers are repunit primes 11111_p, so they are Brazilian primes (A085104).
When p^4 + p^3 + p^2 + p + 1 = sigma(p^4) is prime, then it equals A193574(p^4), so that this sequence is a subsequence of A193574; by definition it is also a subsequence of A053699 and A131992. - Hartmut F. W. Hoft, May 05 2017

			a(3) = 30941 = 11111_13 = 13^4 + 13^3 + 13^2 + 13^1 + 1 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1100

Cf. A049409 (n^4 + ... + 1 is prime), A065509 (primes among these n), A193574.
Subsequence of A088548 (primes n^4 + ... + 1) and A085104 ("Brazilian" primes, of the form 1 + n + n^2 + ... + n^k).
Intersection of A000040 (primes) and A131992 (p^4 + ... + 1), subsequence of A053699 (n^4 + ... + 1).

[p: p in PrimesUpTo(600) | IsPrime(p) where p is p^4 +p^3+p^2+p+1]; // Vincenzo Librandi, May 06 2017

a190527[n_] := Select[Map[(Prime[#]^5-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a190527[100] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[#^4 + #^3 + #^2 + # + 1 &/@Prime[Range[100]], PrimeQ] (* Vincenzo Librandi, May 06 2017 *)

[q|p<-primes(100),ispseudoprime(q=(p^5-1)\(p-1))]
A190527_vec(N)=[(p^5-1)\(p-1)|p<-A065509_vec(N)] \\ M. F. Hasler, Mar 03 2020

a(n) = A193574(A065509(n)^4). - Hartmut F. W. Hoft, May 08 2017

a(n) = A053699(A065509(n)) = A000203(A065509(n)^4). - M. F. Hasler, Mar 03 2020

a(7) corrected and a(18)-a(26) added by Hartmut F. W. Hoft, May 05 2017

Edited by M. F. Hasler, Mar 06 2020

A197654 Triangle by rows T(n,k), showing the number of meanders with length 5(n+1) and containing 5(k+1) L's and 5(n-k) R's, where L's and R's denote arcs of equal length and a central angle of 72 degrees which are positively or negatively oriented.

Original entry on oeis.org

1, 5, 1, 31, 62, 1, 121, 1215, 363, 1, 341, 13504, 20256, 1364, 1, 781, 96875, 500000, 193750, 3905, 1, 1555, 501066, 7321875, 9762500, 1252665, 9330, 1, 2801, 2033647, 72656661, 262609375, 121094435, 6100941, 19607, 1

Offset: 0

Susanne Wienand, Oct 19 2011

Definition of a meander:
A binary curve C is a triple (m, S, dir) such that:
(a) S is a list with values in {L,R} which starts with an L,
(b) dir is a list of m different values, each value of S being allocated a value of dir,
(c) consecutive L's increment the index of dir,
(d) consecutive R's decrement the index of dir,
(e) the integer m>0 divides the length of S.
Then C is a meander if each value of dir occurs length(S)/m times.
Let T(m,n,k) = number of meanders (m, S, dir) in which S contains m(k+1) L's and m(n-k) R's, so that length(S) = m(n+1).
For this sequence, m = 5, T(n,k) = T(5,n,k).
The values in the triangle were proved by brute force for 0 <= n <= 6. The formulas have not yet been proved in general.
The number triangle can be calculated recursively by the number triangles and A007318, A103371, A194595 and A197653. The first column seems to be A053699.  The diagonal right hand is A000012. The diagonal with k = n-1 seems to be A152031 and to start with the second number of A152031. Row sums are in A198257.
The conjectured formulas are confirmed by dynamic programming for 0 <= n <= 29. - Susanne Wienand, Jul 01 2015

			For n = 5 and k = 2, T(n,k) = 500000
Example for recursive formula:
T(1,5,2) = 10
T(4,5,5-1-2) = T(4,5,2) = 40000
T(5,5,2) = 10^5 + 10*40000 = 500000
Example for closed formula:
T(5,2) = A + B + C + D + E
A = 10^5
B = 10^4 * 10
C = 10^3 * 10^2
D = 10^2 * 10^3
E = 10   * 10^4
T(5,2) = 5 * 10^5 = 500000
Some examples of list S and allocated values of dir if n = 5 and k = 2:
Length(S) = (5+1)*5 = 30 and S contains (2+1)*5 = 15 Ls.
  S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R
dir: 1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,0,4,3,2,1,0,4,3,2,1,0,4,3,2,1
  S: L,L,L,L,L,L,L,R,R,L,L,R,R,R,L,R,R,R,L,R,L,L,L,R,R,L,R,R,R,R
dir: 1,2,3,4,0,1,2,2,1,1,2,2,1,0,0,0,4,3,3,3,3,4,0,0,4,4,4,3,2,1
  S: L,L,L,L,L,R,L,L,L,L,L,R,L,R,R,R,R,R,R,R,R,R,R,L,L,L,L,R,R,R
dir: 1,2,3,4,0,0,0,1,2,3,4,4,4,4,3,2,1,0,4,3,2,1,0,0,1,2,3,3,2,1
Each value of dir occurs 30/5 = 6 times.
The triangle begins:
1,
5, 1,
31, 62, 1,
121, 1215, 363, 1,
341, 13504, 20256, 1364, 1,
781, 96875, 500000, 193750, 3905, 1,
...

Alois P. Heinz, Rows n = 0..140, flattened
Peter Luschny, Meanders and walks on the circle.
Susanne Wienand, Example of a meander counted by A197654

Cf. A000012, A007318, A053699, A103371, A152031, A194595, A197653, A197655, A198257.

A197654 := (n,k)->(k^4+2*k^3*(1-n)+2*k^2*(2+n+2*n^2)+k*(3+n-n^2-3*n^3)+ n^4+n^3+n^2+n+1)*binomial(n,k)^5/(1+k)^4;
seq(print(seq(A197654(n, k), k=0..n)), n=0..7);  # Peter Luschny, Oct 20 2011

T[n_, k_] := (k^4 + 2*k^3*(1-n) + 2*k^2*(2+n+2*n^2) + k*(3+n-n^2-3*n^3) + n^4+n^3+n^2+n+1)*Binomial[n, k]^5/(1+k)^4; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 20 2017, after Peter Luschny *)

A197654(n,k) = {if(n ==1+2*k,5,(1+k)*(1-((n-k)/(1+k))^5)/(1+2*k-n))*binomial(n,k)^5} \\ Peter Luschny, Nov 24 2011

def S(N,n,k) : return binomial(n,k)^(N+1)*sum(sum((-1)^(N-j+i)*binomial(N-i,j)*((n+1)/(k+1))^j for i in (0..N) for j in (0..N)))
def A197654(n,k) : return S(4,n,k)
for n in (0..5) : print([A197654(n,k) for k in (0..n)])  # Peter Luschny, Oct 24 2011

Recursive formula (conjectured):
T(n,k) = T(5,n,k) = T(1,n,k)^5 + T(1,n,k)*T(4,n,n-1-k),  0 <= k < n
T(5,n,n) = 1                                             k = n
T(4,n,k) = T(1,n,k)^4 + T(1,n,k) * T(3,n,n-1-k),         0 <= k < n
T(4,n,n) = 1                                             k = n
T(3,n,k) = T(1,n,k)^3 + T(1,n,k) * T(2,n,n-1-k),         0 <= k < n
T(3,n,n) = 1                                             k = n
T(2,n,k) = T(1,n,k)^2 + T(1,n,k) * T(1,n,n-1-k),         0<= k < n
T(2,n,n) = 1                                             k = n
T(4,n,k) = A197653
T(3,n,k) = A194595
T(2,n,k) = A103371
T(1,n,k) = A007318 (Pascal's Triangle)
Closed formula (conjectured): T(n,n) = 1,                              k = n
                T(n,k) = A + B + C + D + E,              k < n
                     A = (C(n,k))^5
                     B = (C(n,k))^4 * C(n,n-1-k)
                     C = (C(n,k))^3 *(C(n,n-1-k))^2
                     D = (C(n,k))^2 *(C(n,n-1-k))^3
                     E =  C(n,k)    *(C(n,n-1-k))^4     [Susanne Wienand]
Let S(n,k) = binomial(2*n,n)^(k+1)*((n+1)^(k+1)-n^(k+1))/(n+1)^k. Then T(2*n,n) = S(n,4). [Peter Luschny, Oct 20 2011]
T(n,k) = A198064(n+1,k+1)C(n,k)^5/(k+1)^4. [Peter Luschny, Oct 29 2011]
T(n,k) = h(n,k)*binomial(n,k)^5, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^5)/(1+2*k-n) if 1+2*k-n <> 0 else h(n,k) = 5. [Peter Luschny, Nov 24 2011]

A105312 a(n) = Sum_{j=0..15} n^j.

Original entry on oeis.org

1, 16, 65535, 21523360, 1431655765, 38146972656, 564221981491, 5538821761600, 40210710958665, 231627523606480, 1111111111111111, 4594972986357216, 16807659899548765, 55451384098598320, 167534872139182395, 469172025408063616, 1229782938247303441

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Apr 30 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), A060885 (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), this sequence (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..15]]): n in [0..20]]; // Vincenzo Librandi, May 01 2011 (modified by G. C. Greubel, Apr 14 2019)

a:= n-> add(n^k, k=0..15):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 04 2012

Prepend[Table[Total[n^Range[0,15]],{n,20}],1]  (* Harvey P. Dale, Jan 19 2011 *)

vector(20, n, n--; sum(j=0,15, n^j)) \\ G. C. Greubel, Apr 14 2019

[sum(n^j for j in (0..15)) for n in (0..20)] # G. C. Greubel, Apr 14 2019

a(n) = n^15 + n^14 + n^13 + n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n^1 + 1.

G.f.: (21845*x^14 + 10412160*x^13 + 689427979*x^12 + 12966588160*x^11 + 93207091581*x^10 + 296077418240*x^9 + 446019954555*x^8 + 326065923072*x^7 + 113735241015*x^6 + 17786608768*x^5 + 1095139065*x^4 + 20476160*x^3 + 65399*x^2 +1 )/(x-1)^16. - Colin Barker, Nov 04 2012

More terms from Harvey P. Dale, Jan 19 2011

Name changed by G. C. Greubel, Apr 14 2019

A059839 a(n) = n^8 + n^6 + n^4 + n^2 + 1.

Original entry on oeis.org

1, 5, 341, 7381, 69905, 406901, 1727605, 5884901, 17043521, 43584805, 101010101, 216145205, 432988561, 820586261, 1483357205, 2574332101, 4311810305, 6999978821, 11054078101, 17030739605, 25664160401, 37908820405, 54989488181, 78459301541, 110266749505, 152832422501

Offset: 0

N. J. A. Sloane, Feb 25 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A059830.

Cf. A060884, A053699.

Table[Total[n^(2*Range[4])]+1,{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,5,341,7381,69905,406901,1727605,5884901,17043521},30] (* Harvey P. Dale, Jan 02 2015 *)

a(n) = { my(f=n^2); f^4 + f^3 + f^2 + f + 1 } \\ Harry J. Smith, Jun 29 2009

a(n) = (n^4-n^3+n^2-n+1)*(n^4+n^3+n^2+n+1) = A060884(n)*A053699(n). a(n) = (n^10-1)/(n^2-1), n>1. - Alexander Adamchuk, Apr 13 2006

G.f.: -(5*x^8 +296*x^7 +4492*x^6 +15332*x^5 +15458*x^4 +4408*x^3 +332*x^2 -4*x +1)/ (x-1)^9. - Colin Barker, Nov 05 2012

A152031 a(n) = n^5 + n^4 + n^3 + n^2 + n.

Original entry on oeis.org

0, 5, 62, 363, 1364, 3905, 9330, 19607, 37448, 66429, 111110, 177155, 271452, 402233, 579194, 813615, 1118480, 1508597, 2000718, 2613659, 3368420, 4288305, 5399042, 6728903, 8308824, 10172525, 12356630, 14900787, 17847788, 21243689, 25137930, 29583455

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 20 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Column k=5 of A228275.

Cf. A053699.

a:= n-> `if`(n=1, 5, (n^6-n)/(n-1)):
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 20 2013

lst={};Do[AppendTo[lst,n^5+n^4+n^3+n^2+n],{n,0,5!}];lst
(* Other programs: *)
Table[Total[n^Range@ 5], {n, 0, 31}] (* or *)
CoefficientList[Series[x (5 + 32 x + 66 x^2 + 16 x^3 + x^4)/(x - 1)^6, {x, 0, 31}], x] (* Michael De Vlieger, Apr 05 2017 *)

a(n) = n^5 + n^4 + n^3 + n^2 + n; \\ Joerg Arndt, Sep 03 2013

def a(n): return n**5 + n**4 + n**2 + n # Indranil Ghosh, Apr 05 2017

a <- c(0, 5, 62, 363, 1364, 3905)
for(n in (length(a)+1):40) a[n] <- 6*a[n-1] -15*a[n-2] +20*a[n-3] -15*a[n-4] +6*a[n-5] -a[n-6]
a
[Yosu Yurramendi, Sep 03 2013]

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) n>5, a(0)=0, a(1)=5, a(2)=62, a(3)=363, a(4)=1364, a(5)=3905. [Yosu Yurramendi, Sep 03 2013]
From Indranil Ghosh, Apr 05 2017: (Start)
G.f.: x*(5 + 32x + 66x^2 + 16x^3 + x^4)/(x - 1)^6.
E.g.f.: exp(x)*x*(5 + 26x + 32x^2 + 11x^3 + x^4).
(End)
a(n) = n*A053699(n). - Michel Marcus, Apr 05 2017

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