A069477 -id:A069477 - cp's OEIS frontend

A069478 First differences of A069477, successive differences of (n+1)^5 - n^5.

360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240, 3360, 3480, 3600, 3720, 3840, 3960, 4080, 4200, 4320, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280

Offset: 1

Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Apr 11 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A022521, A068236, A069477.

[120*(n+2): n in [1..50]]; // G. C. Greubel, Nov 11 2018

Differences[#[[2]]-#[[1]]&/@Partition[Range[50]^5,2,1],3] (* Harvey P. Dale, Jan 28 2013 *)
120(Range[50] +2) (* G. C. Greubel, Nov 11 2018 *)

vector(50, n, 120*(n+2)) \\ G. C. Greubel, Nov 11 2018

a(n) = 120*(n+2).
From Chai Wah Wu, Nov 11 2018: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 2.
G.f.: x*(-240*x + 360)/(x - 1)^2. (End)
E.g.f.: 120*(-2 + (2+x)*exp(x)). - G. C. Greubel, Nov 11 2018

Corrected by T. D. Noe, Mar 14 2008

A101096 Third differences of fifth powers (A000584).

Original entry on oeis.org

1, 29, 150, 390, 750, 1230, 1830, 2550, 3390, 4350, 5430, 6630, 7950, 9390, 10950, 12630, 14430, 16350, 18390, 20550, 22830, 25230, 27750, 30390, 33150, 36030, 39030, 42150, 45390, 48750, 52230, 55830, 59550, 63390, 67350, 71430, 75630, 79950, 84390, 88950

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original Name: Shells (nexus numbers) of shells of shells of the power of 5.

For n>=3 a(n) is equal to the number of functions f:{1,2,3,4,5}->{1,2,...,n} such that Im(f) contains 3 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
David J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al.), National Center for Mathematics Education, University of Gothenburg, Sweden, 2003.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Archive machine link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883.
Eric Weisstein, Link to section of MathWorld: Eulerian Number.
Eric Weisstein, Link to section of MathWorld: Nexus number.
Eric Weisstein, Link to section of MathWorld: Finite Differences.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A069477.
Third differences of A000584, second differences of A022521, and first differences of A068236.
Cf. A101095 for other sequences related to MagicNKZ.
Cf. A001844.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(x^4+26*x^3+66*x^2+26*x+1)/(1-x)^3)); // G. C. Greubel, Dec 01 2018

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 5, 5}, {z, 3, 3}, {k, 0, 34}]
CoefficientList[Series[(-z^4-26z^3-66z^2-26z-1)/(z-1)^3, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Join[{1,29},Differences[Range[0,40]^5,3]] (* or *) LinearRecurrence[{3,-3,1},{1,29,150,390,750},40] (* Harvey P. Dale, Feb 02 2017 *)

a(n)=if(n>2,60*n^2-180*n+150,28*n-27) \\ Charles R Greathouse IV, Oct 11 2015

[sum([(-1)^j*binomial(3, j)*(k-j+1)^5 for j in range(min(k+2,4))]) for k in range(40)] # Danny Rorabaugh, Apr 27 2015

a(k+1) = MagicNKZ(5,k,3) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n. (Cf. A101095.)
a(n+1) = 30*(1 - 2*n + 2*n^2) for n>2.
a(n+3) = A069477(n). - Vladimir Joseph Stephan Orlovsky, Jun 19 2011
G.f.: x*(x^4+26*x^3+66*x^2+26*x+1)/(1-x)^3. - Colin Barker, Oct 17 2012
Sum_{n>=1} 1/a(n) = (Pi/60)*tanh(Pi/2) + 871/870. - Amiram Eldar, Jan 27 2022

MagicNKZ material edited and SeriesAtLevelR material removed by Danny Rorabaugh, Apr 27 2015

cp's OEIS Frontend

A069478 First differences of A069477, successive differences of (n+1)^5 - n^5.

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