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 1-4 of 4 results.

A137361 a(n) = Sum_{k=0..n/2} k*binomial(n-2*k, 3*k+2).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 254, 480, 882, 1617, 2992, 5580, 10410, 19292, 35400, 64343, 116128, 208701, 374226, 670095, 1198164, 2138423, 3808148, 6766089, 11996042, 21229790, 37513896, 66202347, 116692472, 205458357, 361349662, 634845141, 1114205988
Offset: 0

Views

Author

Don Knuth, Apr 11 2008

Keywords

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A137356-A137360, A136444.

Programs

  • Magma
    [&+[k*Binomial(n-2*k, 3*k+2): k in [0..(n div 2)]]: n in [0..40]]; // Bruno Berselli, Feb 13 2015
  • Maple
    a:= n-> (Matrix(10, (i,j)-> if i=j-1 then 1 elif j=1 then [6, -15, 20, -15, 8, -7, 6, -2, 0, -1][i] else 0 fi)^n)[1,8]:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 23 2008
  • Mathematica
    t[i_, j_] := If[i == j-1, 1, If[j == 1, {6, -15, 20, -15, 8, -7, 6, -2, 0, -1}[[i]] , 0]]; M = Array[t, {10, 10}]; a[n_] := MatrixPower[M, n][[1, 8]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Formula

G.f.: x^7/(x^5 + x^3 - 3*x^2 + 3*x - 1)^2. - Alois P. Heinz, Oct 23 2008

A137357 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 12, 23, 44, 80, 138, 230, 379, 629, 1060, 1810, 3109, 5336, 9120, 15521, 26349, 44713, 75949, 129177, 219918, 374521, 637699, 1085401, 1846804, 3141826, 5344988, 9093989, 15474230, 26332515, 44810670, 76253683, 129755543, 220790480
Offset: 0

Views

Author

Don Knuth, Apr 11 2008

Keywords

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A137356, A137358, A137359, A137360, A137361, A136444.

Programs

  • Mathematica
    LinearRecurrence[{3, -3, 1, 0, 1}, Range[0, 4], 50] (* Paolo Xausa, Mar 17 2024 *)

Formula

G.f.: x*(x-1) / (x^5+x^3-3*x^2+3*x-1). - Colin Barker, Jan 23 2013

A137358 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+2).

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 22, 34, 57, 101, 181, 319, 549, 928, 1557, 2617, 4427, 7536, 12872, 21992, 37513, 63862, 108575, 184524, 313701, 533619, 908140, 1545839, 2631240, 4478044, 7619870, 12964858, 22058847, 37533077, 63865592, 108676262, 184929945, 314685488
Offset: 0

Views

Author

Don Knuth, Apr 11 2008

Keywords

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A137356, A137357, A137359, A137360, A137361, A136444.

Programs

  • Mathematica
    Table[Sum[Binomial[n-2k,3k+2],{k,0,Floor[n/2]}],{n,0,50}] (* or *) LinearRecurrence[{3,-3,1,0,1},{0,0,1,3,6},50] (* Harvey P. Dale, Nov 06 2012 *)

Formula

a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(n)=3*a(n-1)-3*a(n-2)+ a(n-3)+ a(n-5). - Harvey P. Dale, Nov 06 2012
G.f.: -x^2/(x^5+x^3-3*x^2+3*x-1). - Colin Barker, Jan 23 2013

A138779 Triangle read by rows: T(n,k)=k*binomial(n-2k,3k+1) (n>=6, 0<=k<=(n-1)/5).

Original entry on oeis.org

1, 5, 15, 35, 70, 126, 2, 210, 16, 330, 72, 495, 240, 715, 660, 1001, 1584, 3, 1365, 3432, 33, 1820, 6864, 198, 2380, 12870, 858, 3060, 22880, 3003, 3876, 38896, 9009, 4, 4845, 63648, 24024, 56, 5985, 100776, 58344, 420
Offset: 6

Views

Author

Emeric Deutsch, May 10 2008

Keywords

Comments

Row n contains floor((n-1)/5) terms.
Row sums yield A137360.

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Crossrefs

Cf. A137360.

Programs

  • Maple
    T:=proc(n,k) options operator, arrow: k*binomial(n-2*k,3*k+1) end proc: for n from 6 to 23 do seq(T(n,k),k=1..(n-1)*1/5) end do; # yields sequence in triangular form
  • Mathematica
    Select[Flatten[Table[k*Binomial[n-2k,3k+1],{n,6,30},{k,0,(n-1)/5}]], #>0&] (* Harvey P. Dale, May 24 2015 *)
Showing 1-4 of 4 results.

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