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.

Previous Showing 11-15 of 15 results.

A300865 Signed recurrence over binary enriched p-trees: a(n) = (-1)^(n-1) + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 2, 4, 6, 10, 16, 27, 46, 77, 131, 224, 391, 672, 1180, 2050, 3626, 6344, 11276, 19863, 35479, 62828, 112685, 200462, 360627, 644199, 1162296, 2083572, 3768866, 6777314, 12289160, 22158106, 40255496, 72765144, 132453122, 239936528, 437445448
Offset: 1

Views

Author

Gus Wiseman, Mar 13 2018

Keywords

Crossrefs

Cf. A000992, A001190, A007317, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300352, A300442, A300443, A300862, A300863, A300864, A300866.

Programs

  • Mathematica
    a[n_]:=a[n]=(-1)^(n-1)+Sum[a[k]*a[n-k],{k,1,n/2}];
Array[a,50]

A099324 Expansion of (1 + sqrt(1 + 4x))/(2(1 + x)).

Original entry on oeis.org

1, 0, -1, 3, -8, 22, -64, 196, -625, 2055, -6917, 23713, -82499, 290511, -1033411, 3707851, -13402696, 48760366, -178405156, 656043856, -2423307046, 8987427466, -33453694486, 124936258126, -467995871776, 1757900019100, -6619846420552, 24987199492704, -94520750408708
Offset: 0

Views

Author

Paul Barry, Oct 12 2004

Keywords

Comments

Binomial transform is A099323. Second binomial transform is A072100.
Hankel transform is A049347. - Paul Barry, Aug 10 2009

Links

Crossrefs

Cf. A014138.

Programs

  • Maple
    f:= gfun:-rectoproc({(2+4*n)*a(n)+(4+5*n)*a(n+1)+(n+2)*a(n+2), a(0) = 1, a(1) = 0}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Mar 27 2018
  • Mathematica
    CoefficientList[Series[(1+Sqrt[1+4x])/(2(1+x)),{x,0,40}],x] (* Harvey P. Dale, Jan 30 2014 *)

Formula

a(n) = Sum_{k=0..2n} (2*0^(2n-k)-1)*C(k,floor(k/2)). - Paul Barry, Aug 10 2009
|a(n+2)| = A091491(n+2,2). - Philippe Deléham, Nov 25 2009
G.f.: T(0)/(2+2*x), where T(k) = k+2 - 2*x*(2*k+1) + 2*x*(k+2)*(2*k+3)/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2013
D-finite with recurrence: (2+4*n)*a(n) + (4+5*n)*a(n+1) + (n+2)*a(n+2) = 0. - Robert Israel, Mar 27 2018

A171368 Another version of A126216.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 5, 0, 9, 0, 1, 0, 0, 0, 21, 0, 14, 0, 1, 0, 0, 14, 0, 56, 0, 20, 0, 1, 0, 0, 0, 84, 0, 120, 0, 27, 0, 1, 0, 0, 42, 0, 300, 0, 225, 0, 35, 0, 1, 0, 0, 0, 330, 0, 825, 0, 385, 0, 44, 0, 1, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 06 2009

Keywords

Comments

Expansion of the first column of the triangle T_(0,x), T_(x,y) defined in A039599; T_(0,0)= A053121, T_(0,1)= A089942, T_(0,2)= A126093, T_(0,3)= A126970.
T(n,k) is the number of Riordan paths of length n with k horizontal steps. A Riordan path is a Motzkin path with no horizontal steps on the x-axis. - Emanuele Munarini, Oct 14 2023

Examples

			Triangle begins:
  1 ;
  0,0 ;
  1,0,0 ;
  0,1,0,0 ;
  2,0,1,0,0 ;
  0,5,0,1,0,0 ;
  5,0,9,0,1,0,0 ;
  ...

Crossrefs

Cf. A000108, A001003,

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A099323(n+1), A126120(n), A005043(n), A000957(n+1), A117641(n) for x = -1, 0, 1, 2, 3 respectively.

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 30, 9, 1, 126, 140, 60, 12, 1, 462, 630, 350, 100, 15, 1, 1716, 2772, 1890, 700, 150, 18, 1, 6435, 12012, 9702, 4410, 1225, 210, 21, 1, 24310, 51480, 48048, 25872, 8820, 1960, 280, 24, 1
Offset: 0

Views

Author

Philippe Deléham, Dec 19 2009

Keywords

Examples

			Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    35,   30,    9,   1;
   126,  140,   60,  12,   1;
   462,  630,  350, 100,  15,  1;
  1716, 2772, 1890, 700, 150, 18, 1;
  ...

Crossrefs

Cf. A107230, A171651
Cf. A001405, A001700, A005573, A005773, A026378, A099323, A122898, A168491.

Programs

  • Mathematica
    T[n_,k_]:=n!SeriesCoefficient[Exp[2*x]*(BesselI[0,2*x]+BesselI[1,2*x])*x^k / k!,{x,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Dec 23 2023 *)

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively.
Conjectural g.f.: 1/(2*t)*( sqrt( (1 - x*t)/(1 - (4 + x)*t) ) - 1 ) = 1 + (3 + x)*t + (10 + 6*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013
E.g.f. of column k: exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 23 2023

A171839 Equal to A171368*A007318.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 3, 2, 1, 0, 0, 6, 8, 3, 1, 0, 0, 15, 22, 15, 4, 1, 0, 0, 36, 68, 52, 24, 5, 1, 0, 0, 91, 198, 191, 100, 35, 6, 1, 0, 0, 232, 586, 651, 425, 170, 48, 7, 1, 0, 0, 603, 1718, 2203, 1656, 820, 266, 63, 8, 1, 0, 0, 1585, 5047, 7285, 6299, 3591, 1435, 392, 80
Offset: 0

Views

Author

Philippe Deléham, Dec 19 2009

Keywords

Comments

Another version of A114586.

Examples

			Triangle begins : 1 ; 0,0 ; 1,0,0 ; 1,1,0,0 ; 3,2,1,0,0 ; 6,8,3,1,0,0 ; ...

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A099323(n+1), A126120(n), A005043(n), A000957(n+1), A117641(n) for x = -2, -1, 0, 1, 2 respectively.
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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