A122993 Expansion of g.f.: A(x) = Product_{n>=0} 1/( 1 - x/(1-x)^n )^( 1/2^(n+1) ).

1, 1, 2, 5, 14, 43, 145, 538, 2194, 9796, 47635, 250811, 1421509, 8623112, 55693506, 381175374, 2753122695, 20909082797, 166448275680, 1385010594903, 12016912542681, 108481226052096, 1016937780320981, 9882191461530141

Offset

0,3

Links

Table of n, a(n) for n=0..23.

David Callan, A permutation pattern that illustrates the strong law of small numbers, arXiv:1111.6297 [math.CO], 2011.

Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Formula

From Paul D. Hanna, Sep 16 2018: (Start)

G.f.: exp( Sum_{n>=0} -log(1 - x/(1-x)^n) / 2^(n+1) ).

G.f.: exp( Sum_{n>=1} x^n / (n*(2 - 1/(1-x)^n)) ). (End)

Example

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 43*x^5 + 145*x^6 + 538*x^7 + 2194*x^8 + 9796*x^9 + 47635*x^10 + ...
such that
A(x) = (1-x)^(-1/2) * (1 - x/(1-x))^(-1/4) * (1 - x/(1-x)^2)^(-1/8) * (1 - x/(1-x)^3)^(-1/16) * ...
RELATED SERIES.
The logarithm of the g.f. can be expressed as
log(A(x)) = x/(2 - 1/(1-x)) + x^2/(2*(2 - 1/(1-x)^2)) + x^3/(3*(2 - 1/(1-x)^3)) + x^4/(4*(2 - 1/(1-x)^4)) + x^5/(5*(2 - 1/(1-x)^5)) + x^6/(6*(2 - 1/(1-x)^6)) + ...
explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 35*x^4/4 + 131*x^5/5 + 534*x^6/6 + 2381*x^7/7 + 11555*x^8/8 + 60580*x^9/9 + 340813*x^10/10 + ...

Mathematica

terms = 24;
gf = Exp[Sum[-2^(-n-1) Log[1-x/(1-x)^n] + O[x]^terms, {n, 0, 5 terms}]];
CoefficientList[gf, x][[1 ;; terms]] // Round (* Jean-François Alcover, Sep 10 2018 *)

Prog

(PARI) {a(n)=round(polcoeff(prod(i=0, 6*n+10, 1/(1-x/(1-x)^i +x*O(x^n))^(1/2^(i+1))), n))}
(PARI) {a(n)=local(A); if(n<0, 0, A=1+O(x); for(k=1, n, A=truncate(A)+x*O(x^k); A+=substvec(A, [x, y], [x/(1-x*y+O(x^k)), y*(1-x*y)]) -A^2*(1-x)); subst(polcoeff(A, n), y, 1))} /* Michael Somos, Oct 21 2006 */

Crossrefs

Cf. A122992.

Sequence in context: A148333 A271270 A201497 * A137552 A137553 A149881

Adjacent sequences: A122990 A122991 A122992 * A122994 A122995 A122996

Keyword

nonn

Author

Paul D. Hanna, Sep 23 2006

Status

approved