OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1276 (first 51 terms from Brian Drake)
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
FORMULA
G.f. g(x) satisfies: g(x) = 1 + x*g(x)^2+x*g(x)/(1-x^2*g(x)^2).
a(n) = sum(k=0..n, binomial(n+k,n)*sum(j=0..k+n+1, binomial(j,-n-3*k+2*j-2) *(-1)^(n+k-j+1)*binomial(n+k+1,j)))/(n+1). - Vladimir Kruchinin, Oct 11 2011
From Peter Bala, Feb 22 2022: (Start)
G.f. g(x) = (1/x)*series reversion of x*(1 + x)*(1 - x)^2/(1 + x - x^2).
It appears that 1 + x*g'(x)/g(x) = 1 + 2*x + 8*x^2 + 41*x^3 + 220*x^4 + ... is the g.f. of A348474. (End)
EXAMPLE
a(4) = 99 since there are 90 Schroeder paths (A006318) from (0,0) to (4,4) plus DNNEN, DNENN, DENNN, DdNN, DNdN, DNNd, EDNNN, ENDNN and dDNN, where E=(1,0), N=(0,1), D=(3,1) and d=(1,1).
MAPLE
A:=series(RootOf(1+_Z*(x-1)+_Z^2*(x-x^2)+_Z^3*x^2-_Z^4*x^3), x, 21): seq(coeff(A, x, i), i=0..20);
MATHEMATICA
a[n_] := Sum[Binomial[n+k, n] * Sum[Binomial[j, -n - 3k + 2j - 2]* (-1)^(n+k-j+1) * Binomial[n+k+1, j], {j, 0, k+n+1}], {k, 0, n}]/(n+1);
a /@ Range[0, 24] (* Jean-François Alcover, Oct 06 2019, after Vladimir Kruchinin *)
PROG
(Maxima) a(n):=sum(binomial(n+k, n)*sum(binomial(j, -n-3*k+2*j-2)*(-1)^(n+k-j+1) *binomial(n+k+1, j), j, 0, k+n+1), k, 0, n)/(n+1); /* Vladimir Kruchinin, Oct 11 2011 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Brian Drake, Sep 20 2007
STATUS
approved