A213128 -id:A213128 - cp's OEIS frontend

A213127 Polylogarithm li(-n,-1/4) multiplied by (5^(n+1))/4.

1, -1, -3, -1, 69, 455, -1155, -50065, -334155, 4107095, 112058925, 491352575, -24429589275, -535893782425, 606194735325, 249291355871375, 4380933801391125, -56204145098271625, -4031655689182933875

Offset: 0

Stanislav Sykora, Jun 06 2012

sign

See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=4.

			polylog(-5,-1/4)*5^6/4 = 455

Stanislav Sykora, Table of n, a(n) for n = 0..100
OEIS-Wiki, Eulerian polynomials

Cf. A212846, A210246, A212847, A213128 through A213157. A015521.

seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*4^k, k=0..n),n=0..18); # Peter Luschny, Apr 21 2013

a[0] = 1; a[n_] := PolyLog[-n, -1/4] * 5^(n+1)/4;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 29 2018 *)

/* see A212846; run limnpq(nmax,1,4) */

x='x+O('x^66); Vec(serlaplace( 5/(4+exp(5*x)) )) \\ Joerg Arndt, Apr 21 2013

a(n) = sum(k=0, n, k!*(-1)^k*5^(n-k)*stirling(n, k, 2)); \\ Seiichi Manyama, Mar 13 2022

See formula in A212846, setting p=1,q=4
From Peter Bala, Jun 24 2012: (Start)
E.g.f.: A(x) = 5/(4 + exp(5*x)) = 1 - x - 3*x^2/2! - x^3/3! + 69*x^4/4! + ....
The compositional inverse (A(-x) - 1)^(-1) = x + 3*x^2/2 + 13*x^3/3 + 51*x^4/4 + 205*x^5/5 + ... is the logarithmic generating function for A015521.
(End)
G.f.: 1/Q(0), where Q(k) = 1 + x*(k+1)/( 1 - 4*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 17 2013
a(n) = Sum_{k=0..n} k! * (-1)^k * 5^(n-k) * Stirling2(n,k). - Seiichi Manyama, Mar 13 2022
a(n) ~ n! * (log(2) * cos(n*arctan(Pi/(2*log(2)))) - Pi * sin(n*arctan(Pi/(2*log(2))))/2) * 5^(n+1) / (Pi^2 + 4*log(2)^2)^(1 + n/2). - Vaclav Kotesovec, May 17 2022

A015531 Linear 2nd order recurrence: a(n) = 4a(n-1) + 5a(n-2).

Original entry on oeis.org

0, 1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604, 8138021, 40690104, 203450521, 1017252604, 5086263021, 25431315104, 127156575521, 635782877604, 3178914388021, 15894571940104, 79472859700521, 397364298502604, 1986821492513021, 9934107462565104

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct vertices of the complete graph K_6. Example: a(2)=4 because the walks of length 2 between the vertices A and B of the complete graph ABCDEF are: ACB, ADB, AEB and AFB. - Emeric Deutsch, Apr 01 2004
General form: k=5^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-4, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=charpoly(A,1). - Milan Janjic, Jan 27 2010
Pisano period lengths: 1, 2, 6, 2, 2, 6, 6, 4, 18, 2, 10, 6, 4, 6, 6, 8, 16, 18, 18, 2,... - R. J. Mathar, Aug 10 2012
The ratio a(n+1)/a(n) converges to 5 as n approaches infinity. - Felix P. Muga II, Mar 09 2014
For odd n, a(n) is congruent to 1 (mod 10). For even n > 0, a(n) is congruent to 4 (mod 10). - Iain Fox, Dec 30 2017

Iain Fox, Table of n, a(n) for n = 0..1431 (terms 0..1000 from Vincenzo Librandi)
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014.
Index entries for linear recurrences with constant coefficients, signature (4,5).

A083425 shifted right.

Cf. A033115 (partial sums), A213128.

[Round(5^n/6): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(5^n/6), n=0..25); # Mircea Merca, Dec 28 2010

LinearRecurrence[{4,5},{0,1},30] (* Harvey P. Dale, Jul 09 2017 *)

a(n)=5^n\/6 ; \\ Charles R Greathouse IV, Apr 14 2014

first(n) = Vec(x/((1 - 5*x)*(1 + x)) + O(x^n), -n) \\ Iain Fox, Dec 30 2017

[lucas_number1(n,4,-5) for n in range(0, 22)] # Zerinvary Lajos, Apr 23 2009

From Paul Barry, Apr 20 2003: (Start)
a(n) = (5^n -(-1)^n)/6.
G.f.: x/((1-5*x)*(1+x)).
E.g.f.(exp(5*x)-exp(-x))/6. (End) (corrected by M. F. Hasler, Jan 29 2012)
a(n) = Sum_{k=1..n} binomial(n, k)*(-1)^(n+k)*6^(k-1). - Paul Barry, May 13 2003
a(n) = 5^(n-1) - a(n-1). - Emeric Deutsch, Apr 01 2004
a(n) = ((2+sqrt(9))^n - (2-sqrt(9))^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009
a(n) = round(5^n/6). - Mircea Merca, Dec 28 2010
The logarithmic generating function 1/6*log((1+x)/(1-5*x)) = x + 4*x^2/2 + 21*x^3/3 + 104*x^4/4 + ... has compositional inverse 6/(5+exp(-6*x)) - 1, the e.g.f. for a signed version of A213128. - Peter Bala, Jun 24 2012
a(n) = (-1)^(n-1)*Sum_{k=0..(n-1)} A135278(n-1,k)*(-6)^k = (5^n - (-1)^n)/6 = (-1)^(n-1)*Sum_{k=0..(n-1)} (-5)^k. Equals (-1)^(n-1)*Phi(n,-5) when n is an odd prime, where Phi is the cyclotomic polynomial. - Tom Copeland, Apr 14 2014

A213129 Polylogarithm li(-n,-1/6) multiplied by (7^(n+1))/6.

Original entry on oeis.org

1, -1, -5, -13, 115, 2099, 11395, -177373, -5116685, -40481581, 948973795, 36701972867, 375364322515, -12090607539661, -580544884927805, -7188739235243293, 301374306966657715, 17150539711123411859, 246564346727945106595, -12988846468460187345853

Offset: 0

Stanislav Sykora, Jun 06 2012

sign

See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=6.

			polylog(-5,-1/6)*7^6/6 = 2099.

Stanislav Sykora, Table of n, a(n) for n = 0..100
OEIS-Wiki, Eulerian polynomials

Cf. A212846, A210246, A212847, A213127, A213128.

Cf. A213130 through A213157

seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*6^k, k=0..n),n=0..17); # Peter Luschny, Apr 21 2013

Table[If[n == 0, 1, PolyLog[-n, -1/6] 7^(n+1)/6], {n, 0, 19}] (* Jean-François Alcover, Jun 27 2019 *)

/* See A212846; run limnpq(nmax,1,6) */

x='x+O('x^66); Vec(serlaplace( 7/(6+exp(7*x)) )) \\ Joerg Arndt, Apr 21 2013

a(n) = sum(k=0, n, k!*(-1)^k*7^(n-k)*stirling(n, k, 2)); \\ Seiichi Manyama, Mar 13 2022

See formula in A212846, setting p=1,q=6.
E.g.f.: 7/(6+exp(7*x)). [Joerg Arndt, Apr 21 2013]
a(n) = Sum_{k=0..n} k! * (-1)^k * 7^(n-k) * Stirling2(n,k). - Seiichi Manyama, Mar 13 2022

A355373 a(n) = Sum_{k=0..n} k! * (-1)^k * n^(n-k) * Stirling2(n,k).

Original entry on oeis.org

1, -1, 0, 3, 40, 455, 2016, -177373, -11564160, -497664081, -12796467200, 536297904659, 132025634657280, 14907422733429239, 1181852660381503488, 34684559693802943875, -11771644802057621110784, -3553614228958108389522721, -656899368126170250221715456

Offset: 0

Seiichi Manyama, Jun 30 2022

sign

Cf. A212846, A213127, A213128, A213129, A213130, A213131, A213132, A213133.

Cf. A318183, A352074.

a[n_] := Sum[k! * (-1)^k * n^(n - k) * StirlingS2[n, k], {k, 0, n}]; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Jun 30 2022 *)

a(n) = sum(k=0, n, k!*(-1)^k*n^(n-k)*stirling(n, k, 2));

a(n) = n! * [x^n] n/(n - 1 + exp(n*x)) for n > 0.

cp's OEIS Frontend

A213127 Polylogarithm li(-n,-1/4) multiplied by (5^(n+1))/4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A015531 Linear 2nd order recurrence: a(n) = 4*a(n-1) + 5*a(n-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A213129 Polylogarithm li(-n,-1/6) multiplied by (7^(n+1))/6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A355373 a(n) = Sum_{k=0..n} k! * (-1)^k * n^(n-k) * Stirling2(n,k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A015531 Linear 2nd order recurrence: a(n) = 4a(n-1) + 5a(n-2).