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

Author

Cecilia Rossiter, Dec 15 2004

Keywords

• easy
• nonn

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).

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.

Programs

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 *)

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

Extensions

New name from Joerg Arndt, Nov 30 2014

Original Formula edited and Crossrefs table added by Danny Rorabaugh, Apr 22 2015