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 11-14 of 14 results.

A307695 Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).

Original entry on oeis.org

1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348, 3444812388280, 52781007848284, 811510465220920, 12513859077134008, 193460383702061200, 2997463389599395270, 46532910920993515900, 723626591914643806180, 11270311875128088314200
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2019

Keywords

Comments

Let 1/(sqrt(1-c*x)*sqrt(1-d*x)) = Sum_{k>=0} b(k)*x^k.
b(n) = Sum_{k=0..n} c^(n-k) * e^k * binomial(n,k) * binomial(2*k,k) = Sum_{k=0..n} d^(n-k) * (-e)^k * binomial(n,k) * binomial(2*k,k), where e = (d-c)/4.
n*b(n) = (c+d)/2 * (2*n-1) * b(n-1) - c * d * (n-1) * b(n-2) for n > 1.

Links

Crossrefs

Cf. A000984 (c=0,d=4,e=1), A026375 (c=1,d=5,e=1), A081671 (c=2,d=6,e=1), A098409 (c=3,d=7,e=1), A098410 (c=4,d=8,e=1), A104454 (c=5,d=9,e=1).
Cf. A084605 (c=-3,d=5,e=2), A098453 (c=-2,d=6,e=2), A322242 (c=-1,d=7,e=2), A084771 (c=1,d=9,e=2), A248168 (c=3,d=11,e=2).
Cf. A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
Cf. A322244 (c=-5,d=11,e=4), A322248 (c=-3,d=13,e=4).

Programs

  • Mathematica
    a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))
  • PARI
    {a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}
  • PARI
    {a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}

Formula

a(n) = Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k).
a(n) = Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k).
D-finite with recurrence: n*a(n) = 10*(2*n-1)*a(n-1) - 64*(n-1)*a(n-2) for n > 1.
a(n) ~ 2^(4*n+1) / sqrt(3*Pi*n). - Vaclav Kotesovec, Apr 30 2019

A098265 G.f. : 1/(1-2x-23x^2)^(1/2).

Original entry on oeis.org

1, 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357, 232260610177, 1327920945601, 7627285093069, 43787832627493, 252042452907169, 1451244932278129, 8370001674641917, 48303478743113893, 279083099450496961
Offset: 0

Views

Author

Paul Barry, Aug 31 2004

Keywords

Comments

Central coefficient of (1+x+6x^2)^n.

Links

Crossrefs

Cf. A084601, A084603, A084605, A098264.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/Sqrt[1-2*x-23*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
  • PARI
    x='x+O('x^66); Vec(1/(1-2*x-23*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

Formula

E.g.f.: exp(x)*BesselI(0, 2*sqrt(6)x).
a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)*6^k}.
a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)*6^k}.
n*a(n) +(1-2n)*a(n-1) +23(1-n)*a(n-2)=0. (Recurrence (4) in the Noe paper).- Veka Gesell, Jun 26 2012
a(n) ~ sqrt(72+6*sqrt(6))*(1+2*sqrt(6))^n/(12*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

A293171 Triangle read by rows: T(n,k) = number of colored weighted Motzkin paths ending at (n,k).

Original entry on oeis.org

1, 1, 1, 9, 2, 1, 25, 15, 3, 1, 145, 52, 22, 4, 1, 561, 285, 90, 30, 5, 1, 2841, 1206, 495, 140, 39, 6, 1, 12489, 6027, 2261, 791, 203, 49, 7, 1, 60705, 27560, 11452, 3864, 1190, 280, 60, 8, 1, 281185, 134073, 54468, 20076, 6174, 1710, 372, 72, 9, 1, 1353769, 633130, 268845, 99240, 33090, 9372, 2370, 480, 85, 10, 1
Offset: 0

Views

Author

N. J. A. Sloane, Oct 19 2017

Keywords

Examples

			Triangle begins:
1,
1,1,
9,2,1,
25,15,3,1,
145,52,22,4,1,
561,285,90,30,5,1,
...

Links

Crossrefs

First column is A084605, 2nd A098520.

Programs

  • Maple
    A293171 := proc(n,k)
    option remember;
    local b,e,c;
    b := 1; e:= 2; c := e^2 ;
    if k < 0 or k > n then
        0;
    elif k = n then
        1;
    elif k = 0 then
        b*procname(n-1,0)+2*c*procname(n-1,1) ;
    else
        procname(n-1,k-1)+b*procname(n-1,k)+c*procname(n-1,k+1) ;
    end if;
end proc:
seq(seq( A293171(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Oct 27 2017
  • Mathematica
    T[n_, k_] := T[n, k] = Module[{b=1, e=2, c=4}, Which[k<0 || k>n, 0, k==n, 1, k == 0, b*T[n-1, 0] + 2*c*T[n-1, 1], True, T[n-1, k-1] + b*T[n-1, k] + c*T[n-1, k+1]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 19 2019, after R. J. Mathar *)

A272868 Triangle read by rows, T(n,k) = 2^k*GegenbauerC(k,-n,-1/4), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 9, 1, 3, 15, 25, 1, 4, 22, 52, 145, 1, 5, 30, 90, 285, 561, 1, 6, 39, 140, 495, 1206, 2841, 1, 7, 49, 203, 791, 2261, 6027, 12489, 1, 8, 60, 280, 1190, 3864, 11452, 27560, 60705, 1, 9, 72, 372, 1710, 6174, 20076, 54468, 134073, 281185
Offset: 0

Views

Author

Peter Luschny, May 08 2016

Keywords

Examples

			Triangle starts:
                                1;
                              1, 1;
                            1, 2, 9;
                          1, 3, 15, 25;
                       1, 4, 22, 52, 145;
                     1, 5, 30, 90, 285, 561;
                 1, 6, 39, 140, 495, 1206, 2841;
             1, 7, 49, 203, 791, 2261, 6027, 12489;

Crossrefs

Cf. A084605, A091147, A098520.

Programs

  • Maple
    T := (n,k) -> simplify(2^k*GegenbauerC(k, -n, -1/4)):
for n from 0 to 9 do seq(T(n,k), k=0..n) od;
  • Mathematica
    Table[If[n == 0, 1, 2^k GegenbauerC[k, -n, -1/4]], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 08 2016 *)

Formula

T(n,n) = A084605(n).
T(n,n-1) = A098520(n).
T(n+1,n)/(n+1) = A091147(n).
Previous Showing 11-14 of 14 results.

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

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