A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

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

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 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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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