cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A081908 a(n) = 2^n*(n^2 - n + 8)/8.

Original entry on oeis.org

1, 2, 5, 14, 40, 112, 304, 800, 2048, 5120, 12544, 30208, 71680, 167936, 389120, 892928, 2031616, 4587520, 10289152, 22937600, 50855936, 112197632, 246415360, 538968064, 1174405120, 2550136832, 5519704064, 11911823360, 25635586048
Offset: 0

Views

Author

Paul Barry, Mar 31 2003

Keywords

Comments

Binomial transform of A000124 (when this begins 1,1,2,4,7,...).
2nd binomial transform of (1,0,1,0,0,0,...).
Case k=2 where a(n,k) = k^n(n^2 - n + 2k^2)/(2k^2) with g.f. (1 - 2kx + (k^2+1)x^2)/(1-kx)^3.

Links

Crossrefs

Cf. A081909.
Cf. A000124, A000079, A001788, A001787.

Programs

  • Magma
    [2^n*(n^2-n+8)/8: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011
  • Mathematica
    Table[2^n*(n^2-n+8)/8, {n,0,50}] (* or *) LinearRecurrence[{6,-12,8}, {1, 2,5}, 50] (* G. C. Greubel, Oct 17 2018 *)
  • PARI
    a(n)=2^n*(n^2-n+8)/8 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (1 - 4*x + 5*x^2)/(1-2*x)^3.
a(n) = A000079(n) + (A001788(n) - A001787(n))/2. - Paul Barry, May 27 2003
a(n) = Sum_{k=0..n} C(n, k)*(1 + C(k, 2)). - Paul Barry, May 27 2003
E.g.f.: (2 + x^2)*exp(2*x)/2. - G. C. Greubel, Oct 17 2018

A081910 a(n) = 4^n*(n^2-n+32)/32.

Original entry on oeis.org

1, 4, 17, 76, 352, 1664, 7936, 37888, 180224, 851968, 3997696, 18612224, 85983232, 394264576, 1795162112, 8120172544, 36507222016, 163208757248, 725849473024, 3212635537408, 14156212207616, 62122406969344, 271579372060672
Offset: 0

Views

Author

Paul Barry, Mar 31 2003

Keywords

Comments

Binomial transform of A081909 4th binomial transform of (1,0,1,0,0,0,......). Case k=4 where a(n,k)=k^n(n^2-n+2k^2)/(2k^2) with G.f.: (1-2kx+(k^2+1)x^2)/(1-kx)^3.

Links

Crossrefs

Cf. A081911.

Programs

  • Magma
    [4^n*(n^2-n+32)/32: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011
  • Maple
    A081910:=n->4^n*(n^2-n+32)/32; seq(A081910(n), n=0..30); # Wesley Ivan Hurt, Mar 12 2014
  • Mathematica
    Table[(4^n (n^2 - n + 32))/32, {n, 0, 30}] (* or *) LinearRecurrence[{12, -48, 64}, {1, 4, 17}, 30] (* Harvey P. Dale, Jan 18 2014 *)
CoefficientList[Series[(1 - 8 x + 17 x^2)/(1 - 4 x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)
  • PARI
    a(n) = 4^n*(n^2-n+32)/32; \\ Joerg Arndt, Mar 12 2014

Formula

a(n) = 4^n*(n^2-n+32)/32.
G.f.: (1-8*x+17*x^2)/(1-4*x)^3.
a(0)=1, a(1)=4, a(2)=17, a(n)=12*a(n-1)-48*a(n-2)+64*a(n-3). - Harvey P. Dale, Jan 18 2014

A081918 a(0) = 1; a(n) = n^(n-1)(3n-1)/2 (n>0).

Original entry on oeis.org

1, 1, 5, 36, 352, 4375, 66096, 1176490, 24117248, 559607373, 14500000000, 414998793616, 13002646487040, 442663617327139, 16271152851709952, 642244372558593750, 27093655358260903936, 1216529796891671712025
Offset: 0

Views

Author

Paul Barry, Mar 31 2003

Keywords

Comments

Main diagonal of square array T(n,k) where T(n,k)=k^n(n^2-n+2k^2)/(2k^2), in which rows have g.f. (1-2kx+(k^2+1)x^2)/(1-kx)^3.

Links

Crossrefs

Cf. A000124, A081908, A081909, A081910, A081911, A081912.

Formula

a(0)=1, a(n)=a(n)=n^n(n^2-n+2n^2)/(2n^2), n>0.
Showing 1-3 of 3 results.

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