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A101101
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The first summation of row 3 of Euler's triangle - a row that will recursively accumulate to the power of 3.
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3
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1, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
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.
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FORMULA
| a(x) = Sum [Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; n = 3, r = -3, or a(x) = 6; x>2, or a(k) = Sum[(-1)^j*Binomial[n + 1 - z, j]*(k - j + 1)^n, {j, 0, k + 1}]; n = 3, z = 1, or a(k) = 6; k>1
G.f.: F(x)=x*(1+4*x+x^2)/(1-x). - L. Edson Jeffery, Jan 29 2012
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MATHEMATICA
| MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}]; Table[MagicNKZ, {n, 3, 3}, {z, 1, 1}, {k, 0, 34}] OR SeriesAtLevelR = Sum[Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; Table[SeriesAtLevelR, {n, 3, 3}, {r, -3, -3}, {x, 4, 35}]
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CROSSREFS
| Within the "cube" of related sequences with construction based upon MaginNKZ formula, with n downward, k rightward and z backward:
Before: this_sequence, A008458, A003215, A000578, A000537, A024166 or A024166, A101094, A101097, A101102
Above: this_sequence, below: A101104, A101100
Within the "cube" of related sequences with construction based upon SeriesAtLevelR formula, with n downward, x rightward and r backward:
Above: this_sequence, below: A101103, A101096.
Sequence in context: A004553 A074826 A018246 * A046786 A035591 A120206
Adjacent sequences: A101098 A101099 A101100 * A101102 A101103 A101104
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KEYWORD
| easy,nonn,uned
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AUTHOR
| Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
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EXTENSIONS
| I wish the sequence was as interesting as the list of references! - N. J. A. Sloane (njas(AT)research.att.com).
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