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.

A182122 Expansion of exp( arcsinh( 2*x ) ).

Original entry on oeis.org

1, 2, 2, 0, -2, 0, 4, 0, -10, 0, 28, 0, -84, 0, 264, 0, -858, 0, 2860, 0, -9724, 0, 33592, 0, -117572, 0, 416024, 0, -1485800, 0, 5348880, 0, -19389690, 0, 70715340, 0, -259289580, 0, 955277400, 0, -3534526380, 0, 13128240840, 0, -48932534040, 0, 182965127280
Offset: 0

Views

Author

Michael Somos, Apr 13 2012

Keywords

Examples

			G.f. = 1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 - 10*x^8 + 28*x^10 - 84*x^12 + ...

Links

Crossrefs

Cf. A104624.
Cf. A000984, A048990, A078531, A224884, A245112, A245113.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Exp(Argsinh(2*x)))); // G. C. Greubel, Aug 12 2018
  • Maple
    s := proc(n) option remember; `if`(n<2, n+1, -4*(n-2)*s(n-2)/(n+1)) end: A127846 := n -> `if`(n<2,n+1,s(n-1)); seq(A127846(n), n=0..47); # Peter Luschny, Sep 23 2014
  • Mathematica
    CoefficientList[Series[Exp[ArcSinh[2x]],{x,0,50}],x] (* Harvey P. Dale, Aug 18 2012 *)
Table[2 HypergeometricPFQ[{-n+1,2-n},{2},-1],{n,0,46}] (* Peter Luschny, Sep 23 2014 *)
  • PARI
    {a(n) = if( n<2, (n>=0) + (n>0), n = n-2; if( n%2, 0, (-1)^(n/2) * 4 * binomial( n, n/2) / (n + 2)))};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sqrt( 1 + 4*x^2 + x*O(x^n) ) + 2*x, n ) )};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = sqrt( 1 + 4*x * A)); polcoeff( A, n))};
  • Sage
    def A182122(n):
    if n < 2: return n+1
    if n % 2 == 1: return 0
    return (-1)^(n/2-1)*binomial(n,n/2)/(n-1)
[A182122(n) for n in range(47)] # Peter Luschny, Sep 23 2014

Formula

G.f.: 2*x + sqrt( 1 + 4*x^2 ) = 1 / (1 - 2*x / (1 + x / (1 - x / (1 + x / ... )))).
The g.f. A(x) satisfies: A(x) = sqrt(1 + 4*x * A(x)).
a(n) = (-1)^n * A104624(n). Convolution inverse of A104624.
Conjecture : n*(n+1)*a(n) + (n+2)*(n-1)*a(n-1) +4*(n+1)*(n-3)*a(n-2) +4*(n+2)*(n-4)*a(n-3) = 0.- R. J. Mathar, Jul 24 2012
a(n) = 2*hypergeom([-n+1,2-n],[2],-1). - Peter Luschny, Sep 23 2014
0 = a(n)*(+16*a(n+2) + 10*a(n+4)) + a(n+2)*(-2*a(n+2) + a(n+4)) if n>=0. - Michael Somos, Jan 10 2017
a(n+4) = 2 * a(n+2) * (a(n+2) - 8*a(n)) / (a(n+2) + 10*a(n)) if n>=0 is even. - Michael Somos, Jan 10 2017
G.f. A(x) satisfies A(x) = 1/A(-x). - Seiichi Manyama, Jun 20 2025

A377268 G.f. satisfies A(x) = (1 - 9*x*A(x))^(1/3).

Original entry on oeis.org

1, -3, 0, 9, 27, 0, -324, -1215, 0, 18711, 75816, 0, -1301265, -5484996, 0, 100048689, 431943435, 0, -8192222064, -35942240565, 0, 700434986472, 3108770417700, 0, -61805774132388, -276711654879477, 0, 5586291123504300, 25180760594032407, 0, -514555201693265040
Offset: 0

Views

Author

Seiichi Manyama, Oct 22 2024

Keywords

Links

Crossrefs

Cf. A104624.

Programs

  • Mathematica
    A377268[n_] := 9^n*Binomial[(2*n - 4)/3, n]/(n + 1);
Array[A377268, 35, 0] (* Paolo Xausa, Aug 05 2025 *)
  • PARI
    a(n) = 9^n*binomial(2*n/3-4/3, n)/(n+1);

Formula

G.f.: (1/x) * Series_Reversion( x/(1-9*x)^(1/3) ).
a(n) = 9^n * binomial(2*n/3 - 4/3,n)/(n+1).
From Seiichi Manyama, Jun 19 2025: (Start)
G.f. A(x) satisfies A(x) = 1/A(-x/A(x)).
a(3*n+2) = 0 for n >= 0. (End)

A206022 Riordan array (1, x*exp(arcsinh(-2*x))).

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 2, -4, 1, 0, 0, 8, -6, 1, 0, -2, -8, 18, -8, 1, 0, 0, 0, -32, 32, -10, 1, 0, 4, 8, 30, -80, 50, -12, 1, 0, 0, 0, 0, 128, -160, 72, -14, 1, 0, -10, -16, -28, -112, 350, -280, 98, -16, 1, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Feb 02 2012

Keywords

Comments

Riordan array (1, x*(sqrt(1+4x^2)-2x)); inverse of Riordan array (1, x/sqrt(1-4x)), see A205813.
The g.f. for row sums (1,1,-1,-1,3,1,-9,1,27,13,-81,67,243,...) is (1+2*x^2+x*sqrt(1+4*x^2))/(1+3*x^2).
Triangle T(n,k), read by rows, given by (0, -2, 1, -1, 1, -1, 1, -1, 1, -1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Examples

			Triangle begins:
  1
  0,   1
  0,  -2,   1
  0,   2,  -4,   1
  0,   0,   8,  -6,    1,
  0,  -2,  -8,  18,   -8,    1
  0,   0,   0, -32,   32,  -10,     1
  0,   4,   8,  30,  -80,   50,   -12,    1
  0,   0,   0,   0,  128, -160,    72,  -14,    1
  0, -10, -16, -28, -112,  350,  -280,   98,  -16,   1
  0,   0,   0,   0,    0, -512,   768, -448,  128, -18,   1
  0,  28,  40,  54,   96,  420, -1512, 1470, -672, 162, -20, 1

Crossrefs

Cf. A104624 (column k=1).

Formula

T(n,n) = 1, T(n+1,n) = -2n = -A005843(n), T(n+2,n) = 2*n^2 = A001105(n), T(n+3,n) = -A130809(n+1), T(2n,n) = A009117(n), T(2n+3,1) = (-1)^n*2*A000108(n).
T(n,k) = T(n-2,k-2) - 4*T(n-2,k-1), for k >= 2.
Showing 1-3 of 3 results.

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