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.

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A106190 Triangle read by rows: T(n,k) = binomial(2(n-k),n-k)/(1-2(n-k)).

Original entry on oeis.org

1, -2, 1, -2, -2, 1, -4, -2, -2, 1, -10, -4, -2, -2, 1, -28, -10, -4, -2, -2, 1, -84, -28, -10, -4, -2, -2, 1, -264, -84, -28, -10, -4, -2, -2, 1, -858, -264, -84, -28, -10, -4, -2, -2, 1, -2860, -858, -264, -84, -28, -10, -4, -2, -2, 1, -9724, -2860, -858, -264, -84, -28, -10, -4, -2, -2, 1, -33592, -9724, -2860, -858
Offset: 0

Views

Author

Paul Barry, Apr 24 2005

Keywords

Comments

Sequence array for expansion of sqrt(1-4x).
Row sums are A106191. Diagonal sums are A106192. Sequence array for A002420. Inverse of number triangle A106187.
Riordan array (sqrt(1-4x),x).

Examples

			Triangle begins
1;
-2,1;
-2,-2,1;
-4,-2,-2,1;
-10,-4,-2,-2,1;
-28,-10,-4,-2,-2,1;

Programs

  • Mathematica
    T[n_, k_] := Binomial[2(n - k), n - k]/(1 - 2(n - k)); Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* Robert G. Wilson v, Apr 25 2005 *)

Extensions

More terms from Robert G. Wilson v, Apr 25 2005

A106193 Expansion of sqrt(1-4x)/(1-2x^2).

Original entry on oeis.org

1, -2, 0, -8, -10, -44, -104, -352, -1066, -3564, -11856, -40720, -141284, -497464, -1768368, -6343808, -22926426, -83402956, -305142432, -1122083312, -4144811244, -15372407464, -57222156528, -213709942208, -800563540356, -3007228179064, -11325019883616
Offset: 0

Views

Author

Paul Barry, Apr 24 2005

Keywords

Crossrefs

Cf. A106192.

Programs

  • Maple
    with(FormalPowerSeries): # requires Maple 2022
rec:=subs(n=n-1,FindRE(sqrt(1-4*x)/(1-2*x^2),x,r(n))); # yields Mathar's recurrence
a:=gfun:-rectoproc({rec, r(0)=1, r(1)=-2, r(2)=0}, r(n), remember);
seq(a(n), n=0..20); # Georg Fischer, Oct 28 2022
  • Mathematica
    CoefficientList[Series[Sqrt[1-4x]/(1-2x^2),{x,0,30}],x] (* Harvey P. Dale, Mar 31 2015 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2(n-2k), n-2k)/(1-2(n-2k))*2^k.
Recurrence: n*a(n) +2*(-2*n+3)*a(n-1) -2*n*a(n-2) +4*(2*n-3)*a(n-3) = 0. - R. J. Mathar, Feb 20 2015
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