A132277 -id:A132277 - cp's OEIS frontend

A106053 Next-to-central column of triangle in A059317.

0, 0, 1, 2, 8, 22, 72, 218, 691, 2158, 6833, 21612, 68726, 218892, 699197, 2237450, 7174018, 23038582, 74097134, 238625222, 769407486, 2483532218, 8024499657, 25951580444, 83999410292, 272098963300, 882045339733, 2861184745710, 9286923094550, 30161343633746

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

Number of h steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths hhh, hH, Hh, hUD, UhD and UDh we have altogether 8 h-steps. a(n) = Sum_{k=0..n-1} k*A132277(n-1,k). - Emeric Deutsch, Sep 03 2007

Number of paths in the right half-plane from (0,0) to (n-1,1) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because we have hhU, HU, hUh, Uhh, UH, DUU, UDU and UUD. Number of h-steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths from (0,0) to (3,0), namely, hhh, hH, Hh, hUD, UhD and UDh, we have altogether 8 h-steps. a(n) = Sum_{k=0..n-1} k*A132277(n-1,k). - Emeric Deutsch, Sep 03 2007

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
José L. Ramírez, The Pascal Rhombus and the Generalized Grand Motzkin Paths, arXiv:1511.04577 [math.CO], 2015.

Cf. A059317, A128720, A132277.

g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2)): gser:=series(g,z=0,33); seq(coeff(gser,z,n),n=0..29); # Emeric Deutsch, Sep 03 2007
g:=((1-z-z^2)*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2))-1/2: gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=0..30); # Emeric Deutsch, Sep 03 2007

t[0, 0] = t[1, 0] = t[1, 1] = t[1, 2] = 1;
t[n_ /; n >= 0, k_ /; k >= 0] /; k <= 2n := t[n, k] = t[n-1, k] + t[n-1, k-1] + t[n-1, k-2] + t[n-2, k-2];
t[n_, k_] /; n<0 || k<0 || k>2n = 0;
a[n_] := t[n-1, n-2];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Aug 07 2018 *)

G.f.: (1 - z - z^2 - sqrt((1+z-z^2)*(1-3z-z^2)))/(2*sqrt((1+z-z^2)*(1-3z-z^2))). - Emeric Deutsch, Sep 03 2007

G.f.: (1-z-z^2)/(2*sqrt((1+z-z^2)*(1-3z-z^2))) - 1/2. - Emeric Deutsch, Sep 03 2007

A132884 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=(1,0) steps (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 8, 0, 1, 13, 0, 15, 0, 1, 0, 57, 0, 24, 0, 1, 63, 0, 156, 0, 35, 0, 1, 0, 384, 0, 340, 0, 48, 0, 1, 321, 0, 1380, 0, 645, 0, 63, 0, 1, 0, 2505, 0, 3800, 0, 1113, 0, 80, 0, 1, 1683, 0, 11145, 0, 8855, 0, 1792, 0, 99, 0, 1, 0, 16008, 0, 37065, 0, 18368, 0, 2736, 0

Offset: 0

Emeric Deutsch, Sep 03 2007

T(2n,0)=A001850(n) (the central Delannoy numbers); T(2n+1,0)=0. T(2n,1)=0; T(2n-1,1)=A108666(n). T(n,k)=0 if n+k is odd. Row sums yield A059345. See A132277 for the same statistic on paths restricted to the first quadrant.

			Triangle starts:
   1;
   0,  1;
   3,  0,  1;
   0,  8,  0,  1;
  13,  0, 15,  0,  1;
   0, 57,  0, 24,  0,  1;
T(3,1)=8 because we have hH, Hh, hUD, UhD, UDh, hDU, DhU and DUh.

Cf. A001850, A108666, A059345, A132277.

G:=1/sqrt((1-t*z-z^2)^2-4*z^2): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) end do; # yields sequence in triangular form

G.f. = 1/sqrt((1-tz-z^2)^2-4z^2).

cp's OEIS Frontend

A106053 Next-to-central column of triangle in A059317.

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