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A168592 G.f.: exp( Sum_{n>=1} A082758(n)*x^n/n ), where A082758(n) = sum of the squares of the trinomial coefficients in row n of triangle A027907. 8
1, 3, 14, 80, 509, 3459, 24579, 180389, 1356743, 10402493, 81004516, 638886082, 5093081983, 40971735401, 332187974718, 2711668091448, 22267979870143, 183830653156341, 1524747465249750, 12700172705956876, 106187411693668179 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k >= 0 and two kinds of (1,1). - Alois P. Heinz, Oct 07 2015

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: A(x) = (1/x)*Series_Reversion[x*(1-x)^2/((1+x)^2*(1-x+x^2))].

G.f.: A(x) satisfies A(x^2) = M(x)*M(-x), where M(x) is the g.f. of A001006. - Alexander Burstein, Oct 03 2017

G.f.: A(x) satisfies A(x^2) = (1-x - sqrt(1-2*x-3*x^2))*(1+x - sqrt(1+2*x-3*x^2))/(4*x^4). - Paul D. Hanna, Oct 05 2017, concluded from formula of Alexander Burstein.

EXAMPLE

G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 509*x^4 + 3459*x^5 +...

log(A(x)) = 3*x + 19*x^2/2 + 141*x^3/3 + 1107*x^4/4 + 8953*x^5/5 +...+ A082758(n)*x^n/n +...

MAPLE

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,

      add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y)))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015

# second Maple program:

a:= proc(n) option remember; `if`(n<4, [1, 3, 14, 80][n+1],

      ((10*(n+1))*(16*n^3-20*n^2-n-1) *a(n-1)

      +(-944*n^4+2596*n^3-1924*n^2+236*n+30) *a(n-2)

      +(90*(n-2))*(16*n^3-52*n^2+45*n-6) *a(n-3)

      -(81*(2*n-5))*(n-2)*(n-3)*(4*n-1) *a(n-4))/

      ((n+1)*(4*n-5)*(2*n+1)*(n+2)))

    end:

seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015

PROG

(PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, sum(k=0, 2*m, polcoeff((1+x+x^2)^m, k)^2)*x^m/m) +x*O(x^n)), n))}

(PARI) {a(n)=polcoeff(1/x*serreverse(x*(1-x)^2/((1+x)^2*(1-x+x^2)+x*O(x^n))), n)}

CROSSREFS

Cf. A168590, A168593, A082758, A027907, A168595, A218321, A263316.

Sequence in context: A020089 A218677 A027614 * A121873 A107596 A212391

Adjacent sequences:  A168589 A168590 A168591 * A168593 A168594 A168595

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 01 2009

STATUS

approved

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Last modified February 23 00:03 EST 2018. Contains 299472 sequences. (Running on oeis4.)