

A101104


a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.


8



1, 12, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24
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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

Table of n, a(n) for n=1..59.
D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The AbelFauvel 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+1z,j)*(kj+1)^n (cf. A101095). That is, a(k) = Sum_{j=0..k+1} (1)^j*binomial(4, j)*(kj+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)/(1x).  Colin Barker, Apr 16 2012


MATHEMATICA

MagicNKZ = Sum[(1)^j*Binomial[n+1z, j]*(kj+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

z = 0  A000007  A019590  .......MagicNKZ(n,k,0) = A008292(n,k+1) .......
z = 1  A000012  A040000  A101101  thisSeq  A101100  .......  .......
z = 2  A000027  A005408  A008458  A101103  A101095  .......  .......
z = 3  A000217  A000290  A003215  A005914  A101096  .......  .......
z = 4  A000292  A000330  A000578  A005917  A101098  .......  .......
z = 5  A000332  A002415  A000537  A000583  A022521  .......  A255181
z = 6  A000389  A005585  A024166  A000538  A000584  A022522  A255177
z = 7  A000579  A040977  A101094  A101089  A000539  A001014  A022523
z = 8  A000580  A050486  A101097  A101090  A101092  A000540  A001015
z = 9  A000581  A053347  A101102  A101091  A101099  A101093  A000541
Cf. A101095 for an expanded table and more about MagicNKZ.
Sequence in context: A227072 A066458 A246342 * A330212 A114455 A048992
Adjacent sequences: A101101 A101102 A101103 * A101105 A101106 A101107


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



