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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 MATHEMATICA (1/x)*InverseSeries[x*(1 - x)^2/((1 + x)^2*(1 - x + x^2)) + O[x]^30, x] // CoefficientList[#, x]& (* Jean-François Alcover, Jun 09 2018 *) 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: A305128 A027614 A306040 * 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 June 18 20:04 EDT 2021. Contains 345121 sequences. (Running on oeis4.)