|
%I #40 Sep 24 2022 10:14:18
%S 1,1,-1,3,-13,71,-461,3447,-29093,273343,-2829325,31998903,-392743957,
%T 5201061455,-73943424413,1123596277863,-18176728317413,
%U 311951144828863,-5661698774848621,108355864447215063,-2181096921557783605
%N G.f. A(x) satisfies: [x^n] A(x)^n = [x^n] A(x)^(n-1) for n>1 with A(0)=A'(0)=1.
%C After initial term, equals signed A003319 (indecomposable permutations).
%H Seiichi Manyama, <a href="/A158882/b158882.txt">Table of n, a(n) for n = 0..449</a>
%F a(n) = (2-n) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - _Michael Somos_, Jul 23 2011
%F a(n) = (-1)^(n-1)*A003319(n) for n>=1.
%F G.f.: A(x) = 1/[Sum_{n>=0} (-1)^n*n!*x^n].
%F G.f. satisfies: [x^(n+1)] A(x)^n = (-1)^n*n*A075834(n+1) for n>=0.
%F From _Sergei N. Gladkovskii_, Jun 24 2012 to May 26 2013: (Start)
%F Continued fractions:
%F Let A(x) be the g.f., then A(x) = 1-x/U(0), where U(k) = x-1+x*k+(k+2)*x/U(k+1).
%F A(x) = 1/U(0), where U(k) = 1 - x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1)- 1/U(k+1))).
%F G.f.: U(0), where U(k)= 1 + x*(k+1)/(1 + x*(k+1)/U(k+1)).
%F G.f.: 2/(G(0) + 1), where G(k)= 1 - x*(k+1)/(1 - 1/(1 + 1/G(k+1))).
%F G.f.: x*G(0), where G(k)=1/x + 2*k + 1 - (k+1)^2/G(k+1).
%F G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1))). (End)
%e G.f.: A(x) = 1 + x - x^2 + 3*x^3 - 13*x^4 + 71*x^5 - 461*x^6 +-...
%e 1/A(x) = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 +...+ (-1)^n*n!*x^n +...
%e ...
%e Coefficients of powers of g.f. A(x) begin:
%e A^1: 1,1,(-1),3,-13,71,-461,3447,-29093,273343,-2829325,...;
%e A^2: 1,2,(-1),(4),-19,110,-745,5752,-49775,476994,-5016069,...;
%e A^3: 1,3, 0, (4),(-21),129,-910,7242,-64155,626319,-6685548,...;
%e A^4: 1,4, 2, 4, (-21),(136),-996,8152,-73811,733244,-7938186,...;
%e A^5: 1,5, 5, 5, -20, (136),(-1030),8650,-79925,807055,-8854741,...;
%e A^6: 1,6, 9, 8, -18, 132, (-1030),(8856),-83385,855010,-9500385,...;
%e A^7: 1,7,14,14, -14, 126, -1008, (8856),(-84861),882805,-9927890,...;
%e A^8: 1,8,20,24, -6, 120, -972, 8712, (-84861),(894928),-10180120,...;
%e A^9: 1,9,27,39,9,117,-927,8469,-83772,(894928),(-10291986),...;
%e A^10:1,10,35,60,35,122,-875,8160,-81890,885620,(-10291986),...; ...
%e where coefficients [x^n] A(x)^n and [x^n] A(x)^(n-1) are
%e enclosed in parenthesis and equal (-1)^n*n*A075834(n+1):
%e [ -1,4,-21,136,-1030,8856,-84861,894928,-10291986,128165720,...];
%e compare to A075834:
%e [1,1,1,2,7,34,206,1476,12123,111866,1143554,12816572,...]
%e and also to the logarithmic derivative of A075834:
%e [1,1,4,21,136,1030,8856,84861,894928,10291986,128165720,...].
%t b[0] = 0; b[n_] := b[n] = n!-Sum[k!*b[n-k], {k, 1, n-1}]; a[0] = 1; a[n_] := (-1)^(n+1)*b[n]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 07 2014, from 2nd formula *)
%o (PARI) a(n)=polcoeff(1/sum(k=0,n,(-1)^k*k!*x^k +x*O(x^n)),n)
%o (PARI) {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-2))-Vec(Ser(A)^(#A-1)))[ #A]);A[n+1]}
%o (Maxima)
%o G(n,k):=(if n=k then 1 else if k=1 then (-sum(binomial(n-1,k-1)*G(n,k),k,2,n)) else sum(G(i+1,1)*G(n-i-1,k-1),i,0,n-k));
%o makelist(G(n,1),n,1,10); /* _Vladimir Kruchinin_, Mar 07 2014 */
%Y Cf. A003319, A075834, A159311, variant: A158883.
%K sign
%O 0,4
%A _Paul D. Hanna_, Apr 30 2009
|
