A158883 G.f. satisfies: [x^n] A(x)^(n+1) = [x^n] A(x)^n for n>1 with A(0)=A'(0)=1.
1, 1, -2, 9, -56, 425, -3726, 36652, -397440, 4695489, -59941550, 821711605, -12037503384, 187689245588, -3104186515976, 54295661153700, -1001685184237056, 19444296845046033, -396260414466644574, 8460628832978195683, -188898511962856879400
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x - 2*x^2 + 9*x^3 - 56*x^4 + 425*x^5 - 3726*x^6 + ... (d/dx) (x/A(x)) = 1 - 2*x + 9*x^2 - 56*x^3 + 425*x^4 - 3726*x^5 + ... 1/A(x) = 1 - x + 3*x^2 - 14*x^3 + 85*x^4 + ... + (-1)^n*A088716(n)*x^n + ... where a(n) = (-1)^(n-1)*n*A088716(n-1) for n >= 1. ... Coefficients of powers of g.f. A(x) begin: A^1: 1,1,-2,9,-56,425,-3726,36652,-397440,4695489,...; A^2: 1,2,(-3),14,-90,702,-6297,63144,-695886,8334822,...; A^3: 1,3,(-3),(16),-108,870,-7997,81774,-915798,11116902,...; A^4: 1,4,-2,(16),(-115),960,-9050,94368,-1073658,13204560,...; A^5: 1,5,0,15,(-115),(996),-9630,102365,-1182690,14730890,...; A^6: 1,6,3,14,-111,(996),(-9870),106890,-1253466,15804548,...; A^7: 1,7,7,14,-105,973,(-9870),(108816),-1294412,16514162,...; A^8: 1,8,12,16,-98,936,-9704,(108816),(-1312227),16931984,...; A^9: 1,9,18,21,-90,891,-9426,107406,(-1312227),(17116900),...; A^10:1,10,25,30,-80,842,-9075,104980,-1298625,(17116900),...; ... where coefficients [x^n] A(x)^(n+1) and [x^n] A(x)^n are enclosed in parenthesis and equal (n+1)*A158884(n) for n > 1: [ -3,16,-115,996,-9870,108816,-1312227,17116900,...]; compare to A158884: [1,1,-1,4,-23,166,-1410,13602,-145803,1711690,-21785618,...] and also to the logarithmic derivative of A158884: [1,-3,16,-115,996,-9870,108816,-1312227,17116900,...].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..448
Programs
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Maple
b:= proc(n) option remember; `if`(n=0, 1, add( b(j)*b(n-j-1)*(j+1), j=0..n-1)) end: a:= n-> `if`(n=0, 1, -(-1)^n*n*b(n-1)): seq(a(n), n=0..25); # Alois P. Heinz, Feb 18 2020 -
Mathematica
m = 19; A[_] = 1; Do[A[x_] = 1 + x*D[x/A[x], x] + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Feb 18 2020 *) -
Maxima
Composita(n,k,F):=if k=1 then F(n) else sum(F(i+1)*Composita(n-i-1,k-1,F),i,0,n-k); array(a, 10); a[1]:1; af(n):=a[n]; for n:2 thru 10 do a[n]:n*sum(Composita(n-1, k, af)*(-1)^k , k, 1, n-1); makelist(af(n),n,1,10); /* Vladimir Kruchinin, Dec 01 2011 */
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PARI
{a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]);A[n+1]}
Formula
G.f. satisfies: A(x) = 1 + x*(d/dx)(x/A(x)) so that x^2*A'(x) = x*A(x) + A(x)^2 - A(x)^3.
a(n) = (-1)^(n-1)*n*A088716(n-1) for n >= 1.
G.f.: A(x) = 1/(Sum_{n>=0} (-1)^n*A088716(n)*x^n), where g.f. F(x) of A088716 satisfies: F(x) = 1 + x*F(x)*(d/dx)(x*F(x)).
G.f. satisfies: [x^n] A(x)^(n+1) = (n+1)*A158884(n) for n > 1.