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)).

1, 1, 3, 1, 7, 2, 19, 9, 1, 51, 28, 3, 141, 95, 18, 1, 393, 306, 70, 4, 1107, 987, 285, 30, 1, 3139, 3144, 1071, 140, 5, 8953, 9963, 3948, 665, 45, 1, 25653, 31390, 14148, 2856, 245, 6, 73789, 98483, 49815, 11844, 1330, 63, 1, 212941, 307836, 172645, 47160

Offset

0,3

Comments

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.

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Formula

G.f.: 1/sqrt((1+z-t*z^2)*(1-3*z-t*z^2)).

T(n,k) = C(n-k,k)*hypergeom([k-n/2,k-n/2+1/2], [1], 4). - Peter Luschny, Sep 18 2014

Example

T(4,1)=9 because we have hhH, hHh, Hhh, HUD, UDH, UHD, HDU, DUH and DHU.
Triangle starts:
                     1;
                     1;
                 3,      1;
                 7,      2;
            19,      9,      1;
            51,     28,      3;
       141,     95,     18,      1;
       393,    306,     70,      4;
  1107,    987,    285,     30,      1;
  3139,   3144,   1071,    140,      5;

Maple

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
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

Mathematica

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 *)

Crossrefs

Columns k=0..3 give A002426, A109188(n-1), A373651(n-4), A375260(n-6).

Row sums gives A059345.

Cf. A132280.

Sequence in context: A235263 A297171 A297156 * A187818 A365454 A059090

Adjacent sequences: A132882 A132883 A132884 * A132886 A132887 A132888

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 03 2007

Status

approved