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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.

%I A133656 #36 Feb 22 2022 08:28:37
%S A133656 1,2,6,23,99,456,2199,10962,56033,292094,1546885,8299058,45010492,
%T A133656 246377362,1359339710,7551689783,42206697209,237156951618,
%U A133656 1338917298708,7591380528489,43207023511013,246773061257046,1413889039642479,8124356140582768,46807462792903984
%N 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.
%H A133656 Alois P. Heinz, <a href="/A133656/b133656.txt">Table of n, a(n) for n = 0..1276</a> (first 51 terms from Brian Drake)
%H A133656 Brian Drake, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953.
%F A133656 G.f. g(x) satisfies: g(x) = 1 + x*g(x)^2+x*g(x)/(1-x^2*g(x)^2).
%F A133656 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
%F A133656 From _Peter Bala_, Feb 22 2022: (Start)
%F A133656 G.f. g(x) = (1/x)*series reversion of x*(1 + x)*(1 - x)^2/(1 + x - x^2).
%F A133656 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)
%e A133656 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).
%p A133656 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);
%t A133656 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);
%t A133656 a /@ Range[0, 24] (* _Jean-François Alcover_, Oct 06 2019, after _Vladimir Kruchinin_ *)
%o A133656 (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 */
%Y A133656 Cf. A006318, A064641, A052709, A063020, A348474.
%Y A133656 Row sums of A201080.
%K A133656 easy,nonn
%O A133656 0,2
%A A133656 _Brian Drake_, Sep 20 2007