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A185971 Convolution inverse of A001147. 4

%I

%S 1,-1,-2,-10,-74,-706,-8162,-110410,-1708394,-29752066,-576037442,

%T -12277827850,-285764591114,-7213364729026,-196316804255522,

%U -5731249477826890,-178676789473121834,-5925085744543837186

%N Convolution inverse of A001147.

%H G. C. Greubel, <a href="/A185971/b185971.txt">Table of n, a(n) for n = 0..400</a>

%F G.f.: 1 / ( Sum_{k>=0} (2*k-1)!! * x^k ).

%F a(n) = -A000698(n) if n > 0.

%F G.f. A(x) = 1 - x * B(x) * C(x) where B = g.f. for A001147 and C = g.f. for A005416.

%F 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

%F From _Paul D. Hanna_, Mar 03 2012: (Start)

%F G.f. A(x) satisfies:

%F (1) A(x) = 1 - x*A(x)^2 * [d/dx x/A(x)^2].

%F (2) [x^n] A(x)^(2*n-2) = [x^n] A(x)^(2*n-1) for n>=2.

%F (3) [x^n] A(x)^(2*n-1) = -(2*n-1)*A000699(n) for n>=1. (End)

%F 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

%F 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

%F 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

%F 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

%F 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

%F 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

%F 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

%F a(n) ~ -n^n * 2^(n+1/2) / exp(n). - _Vaclav Kotesovec_, Feb 23 2014

%e 1 - x - 2*x^2 - 10*x^3 - 74*x^4 - 706*x^5 - 8162*x^6 - 110410*x^7 - ...

%e From _Paul D. Hanna_, Mar 03 2012: (Start)

%e The coefficients in A(x)^n begin:

%e n=1: [1, -1, -2, -10, -74, -706, -8162, -110410, ...];

%e n=2: [1, -2,(-3),-16, -124, -1224, -14516, -200192, ...];

%e n=3: [1, -3,(-3),-19, -156, -1596, -19412, -272772, ...];

%e n=4: [1, -4, -2,(-20),-175, -1856, -23136, -331008, ...];

%e n=5: [1, -5, 0,(-20),-185, -2031, -25920, -377280, ...];

%e n=6: [1, -6, 3, -20,(-189),-2142, -27951, -413568, ...];

%e n=7: [1, -7, 7, -21,(-189),-2205, -29379, -441519, ...];

%e n=8: [1, -8, 12, -24, -186,(-2232),-30324, -462504, ...];

%e n=9: [1, -9, 18, -30, -180,(-2232),-30882, -477666, ...];

%e n=10:[1, -10, 25, -40, -170, -2212,(-31130),-487960, ...];

%e n=11:[1, -11, 33, -55, -154, -2178,(-31130),-494186, ...]; ...

%e where the coefficients in parenthesis demonstrate the properties:

%e (2) [x^n] A(x)^(2*n-2) = [x^n] A(x)^(2*n-1) for n>=2,

%e (3) [x^n] A(x)^(2*n-1) = -(2*n-1)*A000699(n) for n>=1:

%e A000699 = [1/1, 3/3, 20/5, 189/7, 2232/9, 31130/11, ...].

%e Note: g.f. of A000699, G(x), satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].

%e (End)

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

%o (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))}

%o (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 */

%o (Sage)

%o def A185971_list(len): # len >= 1

%o if len == 1: return [1]

%o T = [0]*(2*len-1); T[1] = 1; R = [1,-1]

%o for n in (1..2*len-3):

%o a,b,c = 1,0,0

%o for k in range(n,-1,-1):

%o r = a-(k+2)*c

%o if k < n : T[k+2] = u;

%o a,b,c = T[k-1],a,b

%o u = r

%o T[1] = u;

%o if is_even(n): R.append(-abs(u))

%o return R

%o A185971_list(18) # _Peter Luschny_, Nov 01 2012

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

%K sign

%O 0,3

%A _Michael Somos_, Feb 08 2011

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Last modified June 14 20:21 EDT 2021. Contains 345038 sequences. (Running on oeis4.)