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A127167 T(3n,2n), where T is the array in A047110.

%I A127167 #18 Apr 21 2025 09:41:04
%S A127167 1,5,73,1348,27811,612728,14103464,334974405,8146520511,201822398131,
%T A127167 5074951075766,129185072614240,3322359273912432,86191998671455630,
%U A127167 2252923797781463037,59273686760263160137,1568440076774389559527,41713725234702452284249
%N A127167 T(3n,2n), where T is the array in A047110.
%H A127167 Alois P. Heinz, <a href="/A127167/b127167.txt">Table of n, a(n) for n = 0..200</a>
%F A127167 G.f.: A(x) where (161051*x^2 +1088*x -64) * A(x)^10 + (4600*x -320) * A(x)^9 + (7276*x -624) * A(x)^8 + (7830*x -512) * A(x)^7 + (6189*x +68) * A(x)^6 + (2177*x + 540) * A(x)^5 + 535*A(x)^4 + 278*A(x)^3 + 84*A(x)^2 + 14*A(x) + 1 = 0. - _Mark van Hoeij_, May 01 2013
%e A127167 T(h,k) counts lattice paths restricted as in A047110 by the line y=2x/3, so that the numbers T(3n,2n) are of interest.
%e A127167 T(3,2) counts these 5 paths: RRRUU, RRURU, URRRU, URURR, UURRR.
%p A127167 T:= proc(h, k) option remember;
%p A127167       `if`([h, k]=[0, 0], 1, `if`(h<0 or k<0, 0, T(h-1, k)+
%p A127167       `if`(3*k>2*h and 3*(k-1)<2*h, 0, T(h, k-1))))
%p A127167     end:
%p A127167 a:= n-> T(3*n, 2*n):
%p A127167 seq (a(n), n=0..30); # _Alois P. Heinz_, Apr 04 2012
%p A127167 series(RootOf((161051*x^2+1088*x-64)*A^10 + (4600*x-320)*A^9 + (7276*x-624)*A^8 + (7830*x-512)*A^7 + (6189*x+68)*A^6 + (2177*x+540)*A^5 +535*A^4 +278*A^3 +84*A^2 +14*A +1, A), x=0, 30); # _Mark van Hoeij_, May 01 2013
%t A127167 T[h_, k_] := T[h, k] = If[{h, k} == {0, 0}, 1, If[h < 0 || k < 0, 0, T[h-1, k] + If[3k > 2h && 3(k-1) < 2h, 0, T[h, k-1]]]];
%t A127167 a[n_] := T[3n, 2n];
%t A127167 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 21 2025, after _Alois P. Heinz_ *)
%Y A127167 Cf. A047110.
%K A127167 nonn
%O A127167 0,2
%A A127167 _Clark Kimberling_, Jan 06 2007
%E A127167 More terms from _Alois P. Heinz_, Apr 04 2012