1,2

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

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.

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

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}]

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.

Sequence in context: A042644 A139997 A098117 this_sequence A142827 A142938 A141910

Adjacent sequences: A101093 A101094 A101095 this_sequence A101097 A101098 A101099

easy,nonn

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004