0,4

After initial term, equals signed A003319 (indecomposable permutations).

a(n) = (-1)^(n-1)*A003319(n) for n>=1.

G.f.: A(x) = 1/[Sum_{n>=0} (-1)^n*n!*x^n].

G.f. satisfies: [x^(n+1)] A(x)^n = (-1)^n*n*A075834(n+1) for n>=0.

G.f.: A(x) = 1 + x - x^2 + 3*x^3 - 13*x^4 + 71*x^5 - 461*x^6 +-...

1/A(x) = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 +...+ (-1)^n*n!*x^n +...

...

Coefficients of powers of g.f. A(x) begin:

A^1: 1,1,(-1),3,-13,71,-461,3447,-29093,273343,-2829325,...;

A^2: 1,2,(-1),(4),-19,110,-745,5752,-49775,476994,-5016069,...;

A^3: 1,3, 0, (4),(-21),129,-910,7242,-64155,626319,-6685548,...;

A^4: 1,4, 2, 4, (-21),(136),-996,8152,-73811,733244,-7938186,...;

A^5: 1,5, 5, 5, -20, (136),(-1030),8650,-79925,807055,-8854741,...;

A^6: 1,6, 9, 8, -18, 132, (-1030),(8856),-83385,855010,-9500385,...;

A^7: 1,7,14,14, -14, 126, -1008, (8856),(-84861),882805,-9927890,...;

A^8: 1,8,20,24, -6, 120, -972, 8712, (-84861),(894928),-10180120,...;

A^9: 1,9,27,39,9,117,-927,8469,-83772,(894928),(-10291986),...;

A^10:1,10,35,60,35,122,-875,8160,-81890,885620,(-10291986),...; ...

where coefficients [x^n] A(x)^n and [x^n] A(x)^(n-1) are

enclosed in parenthesis and equal (-1)^n*n*A075834(n+1):

[ -1,4,-21,136,-1030,8856,-84861,894928,-10291986,128165720,...];

compare to A075834:

[1,1,1,2,7,34,206,1476,12123,111866,1143554,12816572,...]

and also to the logarithmic derivative of A075834:

[1,1,4,21,136,1030,8856,84861,894928,10291986,128165720,...].

(PARI) a(n)=polcoeff(1/sum(k=0, n, (-1)^k*k!*x^k +x*O(x^n)), n)

(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]}

Cf. A003319, A075834, A159311, variant: A158883.

Sequence in context: A122455 A126390 A003319 this_sequence A000261 A111140 A137983

Adjacent sequences: A158879 A158880 A158881 this_sequence A158883 A158884 A158885

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 30 2009