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A090365 Shifts 1 place left under the INVERT transform of the BINOMIAL transform of this sequence. 6
1, 1, 3, 11, 47, 225, 1177, 6625, 39723, 251939, 1681535, 11764185, 86002177, 655305697, 5193232611, 42726002123, 364338045647, 3215471252769, 29331858429241, 276224445794785, 2682395337435723, 26832698102762435, 276221586866499839, 2923468922184615897 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The Hankel transform of this sequence is A000178(n+1); example: det([1,1,3; 1,3,11; 3,11,47]) = 12 . - Philippe Deléham, Mar 02 2005

a(n) appears to be the number of indecomposable permutations (A003319) of [n+1] that avoid both of the dashed patterns 32-41 and 41-32. - David Callan, Aug 27 2014

This is true: A nonempty permutation avoids 32-41 and 41-32 if and only if all its components do so. So if A(x) denotes the gf for indecomposable {32-41,41-32}-avoiders, then F(x):=1/(1-A(x)) is the gf for all {32-41,41-32}-avoiders. From A074664, F(x)=1/x(1-1/B(x)) where B(x) is the ogf for the Bell numbers. Solve for A(x). - David Callan, Jul 21 2017

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..225

FORMULA

G.f.: A(x) satisfies A(x) = 1/(1 - A(x/(1-x))*x/(1-x) ).

a(n) = Sum_{k = 0..n} A085838(n, k) . - Philippe Deléham, Jun 04 2004

G.f.: 1/x-1-1/(B(x)-1) where B(x) = g.f. for A000110 the Bell numbers. - Vladeta Jovovic, Aug 08 2004

a(n)=Sum_{k, 0<=k<=n}A094456(n,k). - Philippe Deléham, Nov 07 2007

G.f.: 1/(1-x/(1-2x/(1-x/(1-3x/(1-x/(1-4x/(1-x/(1-5x/(1-... (continued fraction). [Paul Barry, Feb 25 2010]

G.f.: 1 - x/(G(0)+x) ; G(k)= x - 1 + x*k + x*(x-1+x*k)/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 06 2012

G.f.: 1/x - 1/2 + (x^2-4)/(4*U(0)-2*x^2+8) where U(k)= k*(2*k+3)*x^2 + x - 2 - (2-x+2*k*x)*(2+3*x+2*k*x)*(k+1)*x^2/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 28 2012

G.f.: 1/x+1/(U(0)-1)  where U(k)= -x*k + 1 - x - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 29 2012

G.f.: (1 - U(0))/x -1  where U(k)=  1 - x*(k+2) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012

G.f.: (1 - U(0))/x where U(k)= 1 - x*(k+1)/(1-x/U(k+1)); (continued fraction, 1-step). - Sergei N. Gladkovskii, Nov 11 2012

G.f.: 1/x + 1/( G(0)-1) where G(k) = 1 - x/(1 - x*(2*k+1)/(1 - x/(1 - x*(2*k+2)/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Dec 14 2012

G.f.:1/x + 1/( G(0) - 1 ) where G(k) = 1 - x/(1 - x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 03 2013

G.f.: (1 - Q(0))/x where Q(k)=1 + x/(x*k - 1 )/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Feb 23 2013

G.f.: 1/x-1/x/Q(0), where Q(k)= 1 + x/(1 - x + x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 12 2013

MAPLE

bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:

invtr:= proc(p) local b;

           b:= proc(n) option remember; local i;

                `if`(n<1, 1, add(b(n-i) *p(i-1), i=1..n+1))

               end;

        end:

b:= invtr(bintr(a)):

a:= n-> `if`(n<0, 0, b(n-1)):

seq(a(n), n=0..30);  # Alois P. Heinz, Jun 28 2012

MATHEMATICA

a[n_] := Module[{A, B}, A = 1+x; For[k=1, k <= n, k++, B = (A /. x -> x/(1 - x))/(1-x) + O[x]^n // Normal; A = 1 + x*A*B]; SeriesCoefficient[A, {x, 0, n}]]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 23 2016, adapted from PARI *)

PROG

(PARI) {a(n)=local(A); if(n<0, 0, A=1+x+x*O(x^n); for(k=1, n, B=subst(A, x, x/(1-x))/(1-x)+x*O(x^n); A=1+x*A*B); polcoeff(A, n, x))}

CROSSREFS

Cf. A090366, A090367.

Cf. A074664.

Sequence in context: A174347 A062146 A216947 * A035009 A051296 A030832

Adjacent sequences:  A090362 A090363 A090364 * A090366 A090367 A090368

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 26 2003

STATUS

approved

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Last modified February 20 21:53 EST 2018. Contains 299387 sequences. (Running on oeis4.)