A059398 -id:A059398 - cp's OEIS frontend

A132276 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,k), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) (0<=k<=n).

1, 1, 1, 3, 2, 1, 6, 7, 3, 1, 16, 18, 12, 4, 1, 40, 53, 37, 18, 5, 1, 109, 148, 120, 64, 25, 6, 1, 297, 430, 369, 227, 100, 33, 7, 1, 836, 1244, 1146, 760, 385, 146, 42, 8, 1, 2377, 3656, 3519, 2518, 1391, 606, 203, 52, 9, 1, 6869, 10796, 10839, 8188, 4900, 2346, 903, 272

Offset: 0

Emeric Deutsch, Aug 16 2007, Sep 03 2007

Mirror image of A059397. - Emeric Deutsch, Aug 18 2007
Row sums yield A059398.
Riordan matrix (g(x),x*g(x)), where g(x) = (1-x-x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^2). - Emanuele Munarini, May 05 2011

			T(3,2) = 3 because we have UUh, UhU and hUU.
Triangle begins:
    1;
    1,   1;
    3,   2,   1;
    6,   7,   3,  1;
   16,  18,  12,  4,  1;
   40,  53,  37, 18,  5, 1;
  109, 148, 120, 64, 25, 6, 1;
  ...

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Sheng-Liang Yang and Yuan-Yuan Gao, The Pascal rhombus and Riordan arrays, Fib. Q., 56:4 (2018), 337-347. See Fig. 1.

Cf. A059397, A128720 (the leading diagonal).

Cf. A059398.

g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/z^2: G:=simplify(g/(1-t*z*g)): Gser:=simplify(series(G,z=0,13)): for n from 0 to 10 do P[n]:=sort(coeff(Gser, z,n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form

Flatten[Table[Sum[Binomial[2i+k,i]*(k+1)/(i+k+1)*Sum[Binomial[n-j,2i+k]*Binomial[n-k-2i-j,j],{j,0,n-k-2i}],{i,0,(n-k)/2}],{n,0,15},{k,0,n}]] (* Emanuele Munarini, May 05 2011 *)
c[x_] := (1 - Sqrt[1 - 4*x])/(2*x); g[z_] := c[z^2/(1 - z - z^2)^2]/(1 - z - z^2); G[t_, z_] := g[z]/(1 - t*z*g[z]); CoefficientList[ CoefficientList[Series[G[t, x], {x, 0, 49}, {t, 0, 49}], x], t]//Flatten (* G. C. Greubel, Dec 02 2017 *)

create_list(sum(binomial(2*i+k,i) * (k+1)/(i+k+1) * sum(binomial(n-j,2*i+k) * binomial(n-k-2*i-j,j),j,0,n-k-2*i), i,0,(n-k)/2), n,0,15, k,0,n); /* Emanuele Munarini, May 05 2011 */

T(n,k) = sum(i=0, (n-k)/2, (binomial(2*i+k,i)*(k+1)/(i+k+1))*sum(j=0, n-k-2*i, binomial(n-j,2*i+k)*binomial(n-k-2*i-j,j))) \\ G. C. Greubel, Nov 29 2017

T(n,0) = A128720(n).
G.f.: G(t,z) = g/(1-t*z*g), where g = 1 +z*g +z^2*g +z^2*g^2 or g = c(z^2/(1-z-z^2)^2)/(1-z-z^2), where c(z) = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) + T(n-2,k). - Emeric Deutsch, Aug 18 2007
Column k has g.f. z^k*g^(k+1), where g = 1 +z*g +z^2*g +z^2*g^2 = (1 -z-z^2 -sqrt((1+z-z^2)*(1-3*z-z^2)))/(2*z^2).
T(n,k) = Sum_{i=0..(n-k)/2} (binomial(2*i+k,i)*(k+1)/(i+k+1))*Sum_{j=0..(n-k-2*i)} binomial(n-j,2*i+k)*binomial(n-k-2*i-j,j). - Emanuele Munarini, May 05 2011 [corrected by Jason Yuen, Apr 08 2025]

A059397 Triangle formed by right-bounded rhombus rule, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 6, 1, 4, 12, 18, 16, 1, 5, 18, 37, 53, 40, 1, 6, 25, 64, 120, 148, 109, 1, 7, 33, 100, 227, 369, 430, 297, 1, 8, 42, 146, 385, 760, 1146, 1244, 836, 1, 9, 52, 203, 606, 1391, 2518, 3519, 3656, 2377, 1, 10, 63, 272, 903, 2346, 4900, 8188

Offset: 0

N. J. A. Sloane, Jan 29 2001

T(n,n)=A128720(n). Mirror image of A132276. - Emeric Deutsch, Sep 03 2007

			If triangle is reflected in the vertical axis it looks like this:
1
1 1
3 2 1
6 7 3 1
16 18 12 4 1
and now the rhombus rule is clearly visible (e.g. 18 = 6 + 7 + 3 + 2).

W. Klostermeyer et al., A Pascal rhombus, Fibonacci Quarterly, 35 (1976), 318-328.

A variation on A059317. Row sums give A059398.

Cf. A128720, A132276.

g:=proc(z) options operator, arrow: (1/2-(1/2)*z-(1/2)*z^2-(1/2)*sqrt((1+z-z^2)*(1-3*z-z^2)))/z^2 end proc: G:=simplify(g(t*z)/(1-z*g(t*z))): Gser:=simplify(series(G,z=0,13)): for n from 0 to 10 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 10 do seq(coeff(P[n],t,j),j=0..n) end do; # yields sequence in triangular form - Emeric Deutsch, Sep 03 2007

max = 10; g[z_] := (1 - z - z^2 - Sqrt[(1 + z - z^2)*(1 - 3*z - z^2)])/(2 z^2); s = Series[g[t*z]/(1 - z*g[t*z]), {z, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {z, 0, n}, {t, 0, k}]; t[0, 0] = 1; Table[t[n, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 16 2014, after Emeric Deutsch *)

Each entry is sum of 3 entries above it in previous row and the entry directly above two rows back (provided the entries are properly aligned).

G.f.=G(t,z)=g(tz)/(1-zg(tz)), where g(z)=(1-z-z^2-sqrt((1+z-z^2)(1-3z-z^2)))/(2z^2). - Emeric Deutsch, Sep 03 2007

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001

A132887 Number of symmetric paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0).

Original entry on oeis.org

1, 1, 3, 2, 8, 6, 23, 17, 68, 51, 205, 154, 627, 473, 1937, 1464, 6032, 4568, 18900, 14332, 59519, 45187, 188211, 143024, 597241, 454217, 1900821, 1446604, 6065180, 4618576, 19396027

Offset: 0

Emeric Deutsch, Sep 05 2007

nonn

a(2n+1)=A059398(n); a(2n)=A059398(n-1)+A059398(n). The number of all paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) is A128720(n).

			a(4)=8 because we have hhhh, hHh, HH, hUDh, UDUD, UhhD, UHD and UUDD.

Cf. A128720, A059398.

G:=(2*(1+z+z^2))/(1-3*z^2-z^4+sqrt((1+z^2-z^4)*(1-3*z^2-z^4))): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..30);

G.f.=2(1+z+z^2)/[1-3z^2-z^4+sqrt((1+z^2-z^4)(1-3z^2-z^4))].

D-finite with recurrence (n+2)*a(n) +n*a(n-1) +(-n-6)*a(n-2) -2*n*a(n-3) +7*(-n+2)*a(n-4) +5*(-n+4)*a(n-5) +3*(-n+6)*a(n-6) +2*(n-8)*a(n-7) +(3*n-26)*a(n-8) +(n-8)*a(n-9) +(n-10)*a(n-10)=0. - R. J. Mathar, Oct 08 2016

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A132276 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,k), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) (0<=k<=n).

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A059397 Triangle formed by right-bounded rhombus rule, read by rows.

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A132887 Number of symmetric paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0).

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