A185971 Convolution inverse of A001147.

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

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..400

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

Example

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)

Mathematica

a[n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / (Sum[ (2 k - 1)!! x^k, {k, 0, n}] + O[x]^(n + 1)), n]];

Prog

(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

Crossrefs

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

Sequence in context: A306045 A152408 A046863 * A000698 A301932 A289313

Adjacent sequences: A185968 A185969 A185970 * A185972 A185973 A185974

Keyword

sign

Author

Michael Somos, Feb 08 2011

Status

approved