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A101096
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Shells (nexus numbers) of shells of shells of the power of 5.
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4
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1, 29, 150, 390, 750, 1230, 1830, 2550, 3390, 4350, 5430, 6630, 7950, 9390, 10950, 12630, 14430, 16350, 18390, 20550, 22830, 25230, 27750, 30390, 33150, 36030, 39030, 42150, 45390, 48750, 52230, 55830, 59550, 63390, 67350, 71430, 75630, 79950, 84390, 88950
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| For n>=3 a(n) is equal to the number of functions f:{1,2,3,4,5}->{1,2,...,n} such that Im(f) contains 3 fixed elements. - Aleksandar M. Janjic and Milan R. Janjic (agnus(AT)blic.net), Feb 24 2007
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LINKS
| 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
| a(x) = Sum [Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; n = 5, r = -3, or a(x) = {((-4 + x)*(-3 + x))/2 + 26*((-3 + x)*(-2 + x))/2 + 66*((-2 + x)*(-1 + x))/2 + 26*((-1 + x)*x )/2+ (x*(1 + x))/2}; x>2, or a(x) = {150 - 180*x + 60*x^2}; x>2, or a(x) = {30*(5 + 2*(-3 + x)*x)}; x>2, or a(k) = Sum[(-1)^j*Binomial[n + 1 - z, j]*(k - j + 1)^n, {j, 0, k + 1}]; n = 5, z = 3, or a(k) = {30*(1 - 2*k + 2*k^2)}; x>1, or a(k) = {30 - 60*k + 60*k^2}; x>1, or a(k) = {30*(1 + 2*(-1 + k)*k)}; x>1
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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, 5, 5}, {z, 3, 3}, {k, 0, 34}] OR SeriesAtLevelR = Sum[Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; Table[SeriesAtLevelR, {n, 5, 5}, {r, -3, -3}, {x, 3, 35}]
CoefficientList[Series[(-z^4-26z^3-66z^2-26z-1)/(z-1)^3, {z, 0, 200}], z] (* From Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
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CROSSREFS
| Within the "cube" of related sequences with construction based upon the MaginNKZ formula, with n downward, k rightward and z backward.
Before: A101100, A101095, this_sequence, A101098, A022521, A000584, A000539, A101092, A101099
Above: A000217, A000290, A003215, A005914, this_sequence.
Within the "cube" of related sequences with construction based upon the SeriesAtLevelR formula, with n downward, x rightward and r backward ...
Above: A101101, A101103, this_sequence.
Cf. A069477.
Sequence in context: A139997 A103565 A098117 * A142827 A142938 A141910
Adjacent sequences: A101093 A101094 A101095 * A101097 A101098 A101099
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KEYWORD
| easy,nonn
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AUTHOR
| Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
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