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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Jesse Han

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Jesse Han has authored 3 sequences.

A246472 Number of order-preserving (monotone) functions from the power set of 1 = {0} to the power set of n = {0, ..., n-1}.

Original entry on oeis.org

1, 3, 9, 30, 109, 418, 1650, 6604, 26589, 107274, 432934, 1746484, 7040626, 28362324, 114175812, 459344920, 1847008989, 7423262554, 29822432862, 119766845860, 480833598054, 1929896415484, 7744047734652, 31067665113640, 124613703290994, 499744683756868
Offset: 0

Views

Author

Jesse Han, Aug 27 2014

Keywords

Comments

This is the number of ways to choose a pair of elements (x,y) of P(n) so that x is a subset of y. This also gives the number of covariant functors from P(1) to P(n) viewed as categories.

Crossrefs

Matches A129167 with offset 2 for the first four terms.

Programs

  • Mathematica
    Sum[Binomial[#,i](1+ Sum[Binomial[#,j],{j,i+1,#}]),{i,0,#}]& /@ Range[0,20]
  • PARI
    a(n) = sum(i=0, n, binomial(n,i)*(1+ sum(j = i+1, n, binomial(n,j)))); \\ Michel Marcus, Aug 27 2014

Formula

a(n) = Sum_{i=0..n} (binomial(n,i)*(1 + Sum_{j=i+1..n} binomial(n,j))).
a(n) = 2^(2*n-1) + 2^n - binomial(2*n, n)/2. - Vaclav Kotesovec, Aug 28 2014
n*(n-4)*a(n) +2*(-5*n^2+23*n-15)*a(n-1) +4*(8*n^2-41*n+45)*a(n-2) -16*(2*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 15 2017

A212428 a(n) = 18*n + A000217(n-1).

Original entry on oeis.org

0, 18, 37, 57, 78, 100, 123, 147, 172, 198, 225, 253, 282, 312, 343, 375, 408, 442, 477, 513, 550, 588, 627, 667, 708, 750, 793, 837, 882, 928, 975, 1023, 1072, 1122, 1173, 1225, 1278, 1332, 1387, 1443, 1500, 1558, 1617, 1677, 1738, 1800, 1863, 1927, 1992, 2058
Offset: 0

Views

Author

Jesse Han, May 16 2012

Keywords

Comments

Generalization: T(n,i) = A000217(i-1+n) - A000217(i-1) = i*n + A000217(n-1) (corrected by Zak Seidov, Jun 21 2012); in this case is i=18.
For i = 11..16, Milan Janjic observed that if we define f(n,b,i) = Sum_{k=0..n-b} binomial(n,k)*Stirling1(n-k,b)*Product_{j=0..k-1} (-i - j), then T(n-1,i) = -f(n,n-1,i) for n >= 1.

Links

Crossrefs

Cf. A000217, A000096, A001008, A002805, A055998-A056000, A056115, A056119, A056121, A056126, A051942, A101859, A132754-A132758, A212427.

Programs

  • Magma
    [n*(n+35)/2: n in [0..48]]; // Bruno Berselli, Jun 21 2012
  • Mathematica
    Table[-18 (18 - 1)/2 + (18 + n) (17 + n)/2, {n, 0, 100}]
LinearRecurrence[{3,-3,1},{0,18,37},60] (* Harvey P. Dale, Jun 09 2024 *)
  • PARI
    a(n)=n*(n+35)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = (17+n)*(18+n)/2 - 17*18/2 = 18*n + (n-1)*n/2 = n*(n+35)/2.
G.f.: x*(18-17*x)/(1-x)^3. - Bruno Berselli, Jun 21 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 10 2012
a(n) = 18*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
From Amiram Eldar, Jan 11 2021: (Start)
Sum_{n>=1} 1/a(n) = 2*A001008(35)/(35*A002805(35)) = 54437269998109/229732925058000.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/35 - 102126365345729/2527062175638000. (End)
E.g.f.: exp(x)*x*(36 + x)/2. - Elmo R. Oliveira, Dec 12 2024

A212427 a(n) = 17*n + A000217(n-1).

Original entry on oeis.org

0, 17, 35, 54, 74, 95, 117, 140, 164, 189, 215, 242, 270, 299, 329, 360, 392, 425, 459, 494, 530, 567, 605, 644, 684, 725, 767, 810, 854, 899, 945, 992, 1040, 1089, 1139, 1190, 1242, 1295, 1349, 1404, 1460, 1517, 1575, 1634, 1694, 1755, 1817, 1880, 1944, 2009
Offset: 0

Views

Author

Jesse Han, May 16 2012

Keywords

Comments

Generalization: T(n,i) = A000217(i-1+n) - A000217(i-1) = i*n + A000217(n-1); in this case is i=17. See also the comment in A212428.

Links

Crossrefs

Cf. A000217, A000096, A001008, A002805, A055998-A056000, A056115, A056119, A056121, A056126, A051942, A101859, A132754-A132758, A212428.
For n > 22, T(n,17) matches A074170 but with opposite sign.

Programs

  • Magma
    [n*(n+33)/2: n in [0..49]]; // Bruno Berselli, Jun 22 2012
  • Mathematica
    Table[-17 (17 - 1)/2 + (17 + n) (16 + n)/2, {n, 0, 100}]
  • PARI
    a(n)=n*(n+33)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = (16+n)*(17+n)/2 - 16*17/2 = 17*n + (n-1)*n/2 = n*(n+33)/2.
G.f.: x*(17-16*x)/(1-x)^3. - Bruno Berselli, Jun 22 2012
a(n) = 17*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
From Amiram Eldar, Jan 11 2021: (Start)
Sum_{n>=1} 1/a(n) = 2*A001008(33)/(33*A002805(33)) = 53676090078349/216605329340400.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/33 - 14606816124167/340379803249200. (End)
From Elmo R. Oliveira, Dec 12 2024: (Start)
E.g.f.: exp(x)*x*(34 + x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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