OFFSET
1,2
COMMENTS
Original name: The first summation of row 4 of Euler's triangle - a row that will recursively accumulate to the power of 4.
LINKS
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 (1).
FORMULA
a(k) = MagicNKZ(4,k,1) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n (cf. A101095). That is, a(k) = Sum_{j=0..k+1} (-1)^j*binomial(4, j)*(k-j+1)^4.
a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4. - Joerg Arndt, Nov 30 2014
G.f.: x*(1+11*x+11*x^2+x^3)/(1-x). - Colin Barker, Apr 16 2012
MATHEMATICA
MagicNKZ = Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}]; Table[MagicNKZ, {n, 4, 4}, {z, 1, 1}, {k, 0, 34}]
Join[{1, 12, 23}, LinearRecurrence[{1}, {24}, 56]] (* Ray Chandler, Sep 23 2015 *)
CROSSREFS
For other sequences based upon MagicNKZ(n,k,z):
..... | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 | n = 7
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Cf. A101095 for an expanded table and more about MagicNKZ.
KEYWORD
easy,nonn
AUTHOR
Cecilia Rossiter, Dec 15 2004
EXTENSIONS
New name from Joerg Arndt, Nov 30 2014
Original Formula edited and Crossrefs table added by Danny Rorabaugh, Apr 22 2015
STATUS
approved