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Search: id:A101103
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| A101103 |
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The second summation of row 4 of Euler's triangle - a row that will recursively accumulate to the power of 4. Also the shell (nexus numbers) of the shells of the shells of the power of 4. |
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+0
5
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| 1, 13, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 780, 804
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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For n>=3, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that Im(f) contains 3 fixed elements. - Aleksandar M. Janjic and Milan R. Janjic (agnus(AT)blic.net), Mar 08 2007
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LINKS
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Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
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
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a(x) = Sum [Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; n = 4, r = -3, or a(x) = {-3 + 11*(-2 + x) + 11*(-1 + x) + 2*x}; x>2, or a(x) = {-36 + 24*x}; x>1, or a(k) = Sum[(-1)^j*Binomial[n + 1 - z, j]*(k - j + 1)^n, {j, 0, k + 1}]; n = 4, z = 2, or a(k) = {12*(-1 + 2*k)}; k>1, or a(k) = {-12 + 24*k}; k>1
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MATHEMATICA
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MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}]; Table[MagicNKZ, {n, 4, 4}, {z, 2, 2}, {k, 0, 34}] OR SeriesAtLevelR = Sum[Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; Table[SeriesAtLevelR, {n, 4, 4}, {r, -3, -3}, {x, 3, 35}]
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CROSSREFS
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Within the "cube" of related sequences with construction based upon MaginNKZ formula, with n downward, k rightward and z backward . . . Before: A101104, this_sequence, A005914, A005917, A000583, A000538, A101089, A101090, A101091 Above: A005408, A008458, this_sequence, Below: 101095 Within the "cube" of related sequences with construction based upon SeriesAtLevelR formula, with n downward, x rightward, and r backward . . . Before: A101104, this_sequence, A005914, A005917, A000583, A000538, A101089, A101090, A101091 Above: A101101, this_sequence, Below: A101906.
Adjacent sequences: A101100 A101101 A101102 this_sequence A101104 A101105 A101106
Sequence in context: A135172 A034119 A054285 this_sequence A051865 A081928 A034129
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KEYWORD
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easy,nonn,uned
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AUTHOR
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Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
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