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A101096 Third differences of fifth powers (A000584). 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

D. 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, in press.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]

C. 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).

FORMULA

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(k+1) = 30*(1 - 2*k + 2*k^2); k>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

MATHEMATICA

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 *)

PROG

(Sage) [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

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

(MAGMA) m:=40; R<x>:=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

CROSSREFS

Cf. A069477.

Third differences of A000584, second differences of A022521, and first differences of A068236.

Cf. A101095 for other sequences related to MagicNKZ.

Sequence in context: A139997 A103565 A098117 * A142827 A142938 A264252

Adjacent sequences:  A101093 A101094 A101095 * A101097 A101098 A101099

KEYWORD

easy,nonn

AUTHOR

Cecilia Rossiter, Dec 15 2004

EXTENSIONS

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

STATUS

approved

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)