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A132885 Triangle read by rows: T(n,k) is the number of paths in the right half-plane from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k H=(2,0) steps (0 <= k <= floor(n/2)).

%I A132885 #31 Sep 23 2024 04:24:12
%S A132885 1,1,3,1,7,2,19,9,1,51,28,3,141,95,18,1,393,306,70,4,1107,987,285,30,
%T A132885 1,3139,3144,1071,140,5,8953,9963,3948,665,45,1,25653,31390,14148,
%U A132885 2856,245,6,73789,98483,49815,11844,1330,63,1,212941,307836,172645,47160
%N A132885 Triangle read by rows: T(n,k) is the number of paths in the right half-plane from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k H=(2,0) steps (0 <= k <= floor(n/2)).
%C A132885 Row n has 1+floor(n/2) terms. T(n,0)=A002426(n) (the central trinomial coefficients). T(n,1)=A109188(n-1). Row sums yield A059345. See A132280 for the same statistic on paths restricted to the first quadrant.
%H A132885 G. C. Greubel, <a href="/A132885/b132885.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F A132885 G.f.: 1/sqrt((1+z-t*z^2)*(1-3*z-t*z^2)).
%F A132885 T(n,k) = C(n-k,k)*hypergeom([k-n/2,k-n/2+1/2], [1], 4). - _Peter Luschny_, Sep 18 2014
%e A132885 T(4,1)=9 because we have hhH, hHh, Hhh, HUD, UDH, UHD, HDU, DUH and DHU.
%e A132885 Triangle starts:
%e A132885                      1;
%e A132885                      1;
%e A132885                  3,      1;
%e A132885                  7,      2;
%e A132885             19,      9,      1;
%e A132885             51,     28,      3;
%e A132885        141,     95,     18,      1;
%e A132885        393,    306,     70,      4;
%e A132885   1107,    987,    285,     30,      1;
%e A132885   3139,   3144,   1071,    140,      5;
%p A132885 G:=1/sqrt((1+z-t*z^2)*(1-3*z-t*z^2)): Gser:=simplify(series(G,z=0,18)): for n from 0 to 13 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
%p A132885 A132885 := (n,k) -> binomial(n-k,k)*hypergeom([k-n/2,k-n/2+1/2], [1], 4): seq(print(seq(round(evalf(A132885(n,k))),k=0..iquo(n,2))),n=0..9); # _Peter Luschny_, Sep 18 2014
%t A132885 T[n_, k_] := Binomial[n - k, k]*Hypergeometric2F1[k - n/2, k - n/2 + 1/2, 1, 4]; Table[T[n, k], {n,0,10}, {k, 0, Floor[n/2]}] // Flatten  (* _G. C. Greubel_, Mar 01 2017 *)
%Y A132885 Columns k=0..3 give A002426, A109188(n-1), A373651(n-4), A375260(n-6).
%Y A132885 Row sums gives A059345.
%Y A132885 Cf. A132280.
%K A132885 nonn,tabf
%O A132885 0,3
%A A132885 _Emeric Deutsch_, Sep 03 2007