A133656 Number of below-diagonal paths from (0,0) to (n,n) using steps (1,0), (0,1) and (2k-1,1), k a positive integer.
1, 2, 6, 23, 99, 456, 2199, 10962, 56033, 292094, 1546885, 8299058, 45010492, 246377362, 1359339710, 7551689783, 42206697209, 237156951618, 1338917298708, 7591380528489, 43207023511013, 246773061257046, 1413889039642479, 8124356140582768, 46807462792903984
Offset: 0
Examples
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).
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.
Programs
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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);
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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 *) -
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 */
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)