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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A185971 Convolution inverse of A001147.

Original entry on oeis.org

1, -1, -2, -10, -74, -706, -8162, -110410, -1708394, -29752066, -576037442, -12277827850, -285764591114, -7213364729026, -196316804255522, -5731249477826890, -178676789473121834, -5925085744543837186
Offset: 0

Views

Author

Michael Somos, Feb 08 2011

Keywords

Examples

			1 - x - 2*x^2 - 10*x^3 - 74*x^4 - 706*x^5 - 8162*x^6 - 110410*x^7 - ...
From _Paul D. Hanna_, Mar 03 2012: (Start)
The coefficients in A(x)^n begin:
n=1: [1,  -1, -2, -10,  -74,  -706,  -8162, -110410, ...];
n=2: [1,  -2,(-3),-16, -124, -1224, -14516, -200192, ...];
n=3: [1,  -3,(-3),-19, -156, -1596, -19412, -272772, ...];
n=4: [1,  -4, -2,(-20),-175, -1856, -23136, -331008, ...];
n=5: [1,  -5,  0,(-20),-185, -2031, -25920, -377280, ...];
n=6: [1,  -6,  3, -20,(-189),-2142, -27951, -413568, ...];
n=7: [1,  -7,  7, -21,(-189),-2205, -29379, -441519, ...];
n=8: [1,  -8, 12, -24, -186,(-2232),-30324, -462504, ...];
n=9: [1,  -9, 18, -30, -180,(-2232),-30882, -477666, ...];
n=10:[1, -10, 25, -40, -170, -2212,(-31130),-487960, ...];
n=11:[1, -11, 33, -55, -154, -2178,(-31130),-494186, ...]; ...
where the coefficients in parenthesis demonstrate the properties:
(2) [x^n] A(x)^(2*n-2) = [x^n] A(x)^(2*n-1) for n>=2,
(3) [x^n] A(x)^(2*n-1) = -(2*n-1)*A000699(n) for n>=1:
A000699 = [1/1, 3/3, 20/5, 189/7, 2232/9, 31130/11, ...].
Note: g.f. of A000699, G(x), satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].
(End)

Links

Crossrefs

Cf. A001147, A000699, A185971, A208975, A208961.

Programs

  • Mathematica
    a[n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / (Sum[ (2 k - 1)!! x^k, {k, 0, n}] + O[x]^(n + 1)), n]];
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 / sum( k=0, n, x^k * (2*k)! / (2^k * k!), x * O(x^n)), n))}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1-x*A^2*deriv(x/A^2));polcoeff(A,n)} /* Paul D. Hanna, Mar 03 2012 */
  • Sage
    def A185971_list(len): # len >= 1
    if len == 1: return [1]
    T = [0]*(2*len-1); T[1] = 1; R = [1,-1]
    for n in (1..2*len-3):
        a,b,c = 1,0,0
        for k in range(n,-1,-1):
            r = a-(k+2)*c
            if k < n : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u;
        if is_even(n): R.append(-abs(u))
    return R
A185971_list(18)  # Peter Luschny, Nov 01 2012

Formula

G.f.: 1 / ( Sum_{k>=0} (2*k-1)!! * x^k ).
a(n) = -A000698(n) if n > 0.
G.f. A(x) = 1 - x * B(x) * C(x) where B = g.f. for A001147 and C = g.f. for A005416.
G.f.: A(x) = 1 - x/W(0); W(k) = 1 + x + x*2k - x*(2k+3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
From Paul D. Hanna, Mar 03 2012: (Start)
G.f. A(x) satisfies:
(1) A(x) = 1 - x*A(x)^2 * [d/dx x/A(x)^2].
(2) [x^n] A(x)^(2*n-2) = [x^n] A(x)^(2*n-1) for n>=2.
(3) [x^n] A(x)^(2*n-1) = -(2*n-1)*A000699(n) for n>=1. (End)
G.f. A(x) = G(0) where G(k)= 1 - x*(2*k+1)/(1 - (2*k+2)*x/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 11 2012
G.f. A(x)=1-x/Q(0) where Q(k)= 1 - (k+2)*x/Q(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 20 2012
G.f. A(x) = G(0) where G(k)= 1 - x*(k+1)/G(k+1); (continued fraction, 1-step).- Sergei N. Gladkovskii, Oct 28 2012
G.f.: 1/(1 + x*(Q(0) - 1)/(x+1)) where Q(k)= 1 + (2*k+1)/(1-x/(x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 11 2013
G.f.: Q(0), where Q(k)= 1 + (k+1)*sqrt(x) - sqrt(x)/(1-sqrt(x)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 18 2013
G.f.: Q(0), where Q(k)= 1 + (2*k+1)*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
a(n) ~ -n^n * 2^(n+1/2) / exp(n). - Vaclav Kotesovec, Feb 23 2014

A286794 Row sums of A286781.

Original entry on oeis.org

1, 3, 20, 189, 2232, 31130, 497016, 8907885, 176829104, 3849436062, 91187523000, 2335691914050, 64344487654800, 1897619527612692, 59667237154623280, 1993022006345620605, 70488571028815935072, 2631925423768158446390, 103469607286411235941944, 4272438866376100717458486
Offset: 0

Views

Author

Gheorghe Coserea, May 16 2017

Keywords

Examples

			A(x) = 1 + 3*x + 20*x^2 + 189*x^3 + 2232*x^4 + 31130*x^5 + ...

Links

Crossrefs

Cf. A208975, A286781.

Programs

  • Mathematica
    max = 22; (* B(x) is A000699(x) *) B[_] = 0;
Do[B[x_] = x + x^2 D[B[x]^2/x, x] + O[x]^max // Normal, max];
A[x_] = (1 - x/B[x])/x + O[x]^max;
Drop[CoefficientList[A[x], x], -2] (* Jean-François Alcover, Oct 25 2018 *)
  • PARI
    A286781_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
  while(n++,
    y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
    if (y1 == y0, break()); y0 = y1;);
  y0;
};
Vec(A286781_ser(20,1))
  • PARI
    A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
A286794_seq(N) = Vec((1-1/Ser(A000699_seq(N+1)))/x);
A286794_seq(20)

Formula

a(n) = Sum_{k=0..n} A286781(n,k).
A(x) = (1-x/A000699(x))/x, A208975(x) = 1 + x*A(-x).
a(n) ~ 4*exp(-1)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 3/(8*n) - 215/(128*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 23 2017
Showing 1-2 of 2 results.

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