A128720 -id:A128720 - cp's OEIS frontend

A059345 Central column of Pascal's "rhombus" (actually a triangle) A059317.

1, 1, 4, 9, 29, 82, 255, 773, 2410, 7499, 23575, 74298, 235325, 747407, 2381126, 7603433, 24332595, 78013192, 250540055, 805803691, 2595158718, 8368026845, 27012184877, 87283372610, 282294378071, 913775677281, 2960160734818

Offset: 0

N. J. A. Sloane, Jan 27 2001

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). Example: a(3)=9 because we have hhh, hH, Hh, hUD, hDU, UhD, DhU, UDh and DUh. The number of such paths restricted to the first quadrant is given in A128720. - Emeric Deutsch, Sep 03 2007
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), (2,2). - Joerg Arndt, Jun 30 2011
Other two columns of the triangle in A059317 are given in A106053 and A106050. - Emeric Deutsch, Sep 03 2007

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

T. D. Noe, Table of n, a(n) for n=0..200
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 231-236.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv:1504.04404 [math.CO], 2015.

Cf. A128720, A106050, A106053.

Cf. A181545. - Paul D. Hanna, Oct 29 2010

r:=proc(i,j) if i=0 then 0 elif i=1 and abs(j)>0 then 0 elif i=1 and j=0 then 1 elif i>=1 then r(i-1,j)+r(i-1,j-1)+r(i-1,j+1)+r(i-2,j) else 0 fi end: seq(r(i,0),i=1..12); # very slow; Emeric Deutsch, Jun 06 2004
G:=1/sqrt((1+z-z^2)*(1-3*z-z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z, n),n=0..27); # Emeric Deutsch, Sep 03 2007
a[0]:=1: a[1]:=1: a[2]:=4: a[3]:=9: for n from 3 to 26 do a[n+1]:=((2*n+1)*a[n]+5*n*a[n-1]-(2*n-1)*a[n-2]-(n-1)*a[n-3])/(n+1) end do: seq(a[n],n=0..27); # Emeric Deutsch, Sep 03 2007

CoefficientList[Series[1/Sqrt[(1+x-x^2)(1-3x-x^2)],{x,0,40}],x] (* Harvey P. Dale, Jun 04 2011 *)
a[n_] := Sum[Binomial[n-k, k]*Hypergeometric2F1[(2*k-n)/2, (2*k-n+1)/2, 1, 4], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 26 2015 *)

{a(n)=polcoeff(sum(m=0,n,x^(2*m)/(1-x-x^2+x*O(x^n))^(2*m+1)*(2*m)!/(m!)^2),n)} \\ Paul D. Hanna, Oct 29 2010

/* same as in A092566 but use */
steps=[[1,0], [1,1], [1,2], [2,2]];
/* Joerg Arndt, Jun 30 2011 */

G.f.: 1/sqrt((1+z-z^2)*(1-3*z-z^2)). - Emeric Deutsch, Sep 03 2007
D-finite with recurrence: (n+1)*a(n+1)=(2*n+1)*a(n)+5*n*a(n-1)-(2*n-1)*a(n-2)-(n-1)*a(n-3). - Emeric Deutsch, Sep 03 2007
a(n) = sum{k=0..floor(n/2), C(n-k,k)*A002426(n-2k)}. - Paul Barry, Nov 29 2008
G.f.: A(x) = Sum_{n>=0} (2*n)!/(n!)^2 * x^(2n)/(1-x-x^2)^(2n+1). - Paul D. Hanna, Oct 29 2010
a(n) ~ sqrt((3+11/sqrt(13))/8) * ((3+sqrt(13))/2)^n/sqrt(Pi*n). - Vaclav Kotesovec, Aug 11 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001

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

Original entry on oeis.org

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

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).

Original entry on oeis.org

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]

A059398 Row sums of triangle in A059397.

Original entry on oeis.org

1, 2, 6, 17, 51, 154, 473, 1464, 4568, 14332, 45187, 143024, 454217, 1446604, 4618576, 14777451, 47371177, 152110326, 489165277, 1575211177, 5078690936, 16392526502, 52963765321, 171282782902, 554393341371, 1795821017014

Offset: 0

N. J. A. Sloane, Jan 29 2001

Number of paths in the first quadrant from (0,0) to the line x=n, consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) (in other words, left factors of the paths in A128720). Example: a(2)=6 because we have hh, H, UD, hU, Uh and UU. Row sums of triangle in A132276. - Emeric Deutsch, Sep 03 2007

Row sums of the 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) (A132276). - Emanuele Munarini, May 05 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
W. Klostermeyer et al., A Pascal rhombus, Fibonacci Quarterly, 35 (1976), 318-328.

Cf. A128720, A132276.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!(Sqrt((1+x-x^2)/(1-3*x-x^2))-1)/(2*x)) // G. C. Greubel, Jan 29 2018

g:=(1/2)*(sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x: gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=0..25); # Emeric Deutsch, Sep 03 2007

Table[Sum[Binomial[2k,k](-1)^(n-k+1)Sum[Binomial[i+k-1,i]Binomial[i,n-k-i+1],{i,0,n-k+1}],{k,0,n+1}]/2,{n,0,28}] (* Emanuele Munarini, May 05 2011 *)
With[{nn = 50}, CoefficientList[Series[(Sqrt[(1 + x - x^2)/(1 - 3*x - x^2)] - 1)/x/2, {x, 0, nn}], x]] (* G. C. Greubel, Jan 29 2018 *)

makelist(sum(binomial(2*k,k)*(-1)^(n-k+1)*sum(binomial(i+k-1,i)*binomial(i,n-k-i+1),i,0,n-k+1),k,0,n+1)/2,n,0,28); /* Emanuele Munarini, May 05 2011 */

x='x+O('x^30); Vec((sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x/2) \\ G. C. Greubel, Jan 29 2018

G.f.: (sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x/2. - Vladeta Jovovic, Jan 20 2004

a(n) = (1/2)*sum(binomial(2*k,k)*(-1)^(n-k+1)*sum(binomial(i+k-1,i)*binomial(i,n-k-i+1),i=0..n-k+1),k=0..n+1). - Emanuele Munarini, May 05 2011

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

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

A174168 A (1,3) Somos-4 sequence.

Original entry on oeis.org

1, 2, 5, 17, 109, 706, 9529, 149057, 3464585, 141172802, 5987285341, 540281496721, 53055650250901, 8453268527614082, 2173247123169247025, 610193552206250342657, 364280506115937098013649, 225791960727416355391439746

Offset: 0

Paul Barry, Mar 10 2010

Hankel transform of A128720.

Reinhard Zumkeller, Table of n, a(n) for n = 0..100
Paul Barry, Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences , J. Int. Seq. 13 (2010) #10.7.2.
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490 [math.CO], 2011.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
P. Rajkovic, P. Barry, and N. Savic, Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants, Math. Balkanica, Vol. 26 (2012), Fasc. 1-2.

Cf. A128720.

a174168 n = a174168_list !! n
a174168_list = [1,2,5,17] ++ zipWith div (zipWith (+)
   (zipWith (*) (tail a174168_list) (drop 3 a174168_list))
                (map ((* 3) . (^ 2)) (drop 2 a174168_list))) a174168_list
-- Reinhard Zumkeller, Feb 13 2012

I:=[1,2,5,17]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + 3*Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Sep 25 2018

A174168 := proc(n)
        if n <=4 then
                op(n+1,[1,2,5,17,109]) ;
        else
                (procname(n-1)*procname(n-3)+3*procname(n-2)^2)/procname(n-4) ;
        end if;
end proc: # R. J. Mathar, Oct 26 2011

RecurrenceTable[{a[0]==1,a[1]==2,a[2]==5,a[3]==17,a[n]==(a[n-1]a[n-3]+ 3a[n-2]^2)/a[n-4]},a[n],{n,20}] (* Harvey P. Dale, Oct 25 2011 *)
a[ n_] := Which[n < -1, a[-2 - n], n < 2, {1, 1, 2}[[n + 2]], True, (a[n - 1] a[n - 3] + 3 a[n - 2]^2)/a[n - 4]]; (* Michael Somos, Apr 30 2020 *)

m=20; v=concat([1,2,5,17], vector(m-4)); for(n=5, m, v[n] = ( v[n-1]*v[n-3] +3*v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Sep 25 2018

a(n) = (a(n-1)*a(n-3) + 3*a(n-2)^2)/a(n-4) [index corrected by Franklin T. Adams-Watters, Oct 26 2011].

a(n) = a(-2-n), a(n)*a(n-5) = -3*a(n-1)*a(n-4) + 16*a(n-2)*a(n-3) for all n in Z. - Michael Somos, Apr 30 2020

More terms from Sean A. Irvine, Feb 12 2012

A128719 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UUU's (triplerises) (n >= 0; 0 <= k <= n-2 for n >= 2).

Original entry on oeis.org

1, 1, 3, 6, 4, 16, 12, 8, 40, 53, 28, 16, 109, 176, 162, 64, 32, 297, 625, 633, 456, 144, 64, 836, 2084, 2677, 2024, 1216, 320, 128, 2377, 7016, 10257, 9849, 6008, 3120, 704, 256, 6869, 23218, 39378, 42222, 32930, 16928, 7776, 1536, 512, 20042, 76811, 146191

Offset: 0

Emeric Deutsch, Mar 30 2007

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
Row n has n-1 terms (n >= 2).
Row sums yield A002212.

			T(3,1)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
Triangle starts:
   1;
   1;
   3;
   6,  4;
  16, 12,  8;
  40, 53, 28, 16;

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Cf. A002212, A128720, A128721.

eq:=z*(t+z-t*z)*G^2-(1-z-z^2+t*z^2)*G+1-t*z=0: G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,14)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: 1; 1; for n from 2 to 11 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form

T(n,0) = A128720(n).
Sum_{k=0..n-2} k*T(n,k) = A128721(n) for n >= 2.
G.f.: G = G(t,z) satisfies z(t + z - tz)G^2 - (1 - z - z^2 + tz^2)G + 1 - tz = 0.

A132277 Triangle read by rows: T(n,k) is number of paths in the first quadrant from (0,0) to (n,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having exactly k h-steps.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 5, 0, 1, 6, 0, 9, 0, 1, 0, 25, 0, 14, 0, 1, 22, 0, 66, 0, 20, 0, 1, 0, 129, 0, 140, 0, 27, 0, 1, 90, 0, 450, 0, 260, 0, 35, 0, 1, 0, 681, 0, 1210, 0, 441, 0, 44, 0, 1, 394, 0, 2955, 0, 2765, 0, 700, 0, 54, 0, 1, 0, 3653, 0, 9625, 0, 5642, 0, 1056, 0, 65, 0, 1

Offset: 0

Emeric Deutsch, Aug 26 2007

T(2n+1,0)=0; T(2n,0)=A006318(n) (the large Schroeder numbers). Row sums yield A128720. Sum(k*T(n,k),k=0..n)=A106053(n+1).

			T(4,2)=9 because we have hhH, hhUD, hHh, hUDh, Hhh, UDhh, hUhD, UhDh and UhhD.

Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

Cf. A006318, A128720, A106053.

G:=((1-t*z-z^2-sqrt((1-2*z-t*z-z^2)*(1+2*z-t*z-z^2)))*1/2)/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. G=G(t,z) satisfies G = 1 + tzG + z^2*G + z^2*G^2 (see explicit expression at the Maple program).

A246974 Number of 3-ary plane multitrees with n edges.

Original entry on oeis.org

1, 1, 3, 10, 28, 93, 317, 1090, 3876, 13995, 51182, 189606, 709218, 2675230, 10166639, 38883721, 149559230, 578153160, 2245017535, 8752828951, 34250020397, 134465596581, 529509173245, 2090920335200, 8277633788511, 32846871639751, 130624556118075, 520512049658200

Offset: 0

N. J. A. Sloane, Sep 14 2014

nonn

A k-ary plane multitree is a plane tree with edges having multiplicity and the outdegree of any node does not exceed k. The number of plane multitrees with n edges (without restriction on outdegree) is given by A002212(n). - Andrew Howroyd, Feb 24 2020

Andrew Howroyd, Table of n, a(n) for n = 0..200
M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.

Cf. A002212, A128720 (2-ary case), A246975 (4-ary case).

a(n)={my(m=3); if(n<1, n==0, sum(k=1, n+1, sum(i=1, k-1, sum(j=0, (n-i)\m, (-1)^j*binomial(k, i)*binomial(i, j)*binomial(n-i, k-i-1)*binomial(n-m*j-1, i-1)))/k))} \\ Andrew Howroyd, Feb 24 2020

a(n) = Sum_{k=1..n+1} Sum_{i=1..k-1} Sum_{j=0..floor((n-i)/3)} (-1)^j*binomial(k, i)*binomial(i, j)*binomial(n-i, k-i-1)*binomial(n-3*j-1, i-1)/k for n > 0. - Andrew Howroyd, Feb 24 2020

Terms a(11) and beyond from Andrew Howroyd, Feb 24 2020

A246975 Number of 4-ary plane multitrees with n edges.

Original entry on oeis.org

1, 1, 3, 10, 36, 121, 447, 1699, 6589, 25914, 103633, 419421, 1714463, 7068285, 29361629, 122764876, 516245009, 2181957489, 9264275600, 39495666700, 169000837410, 725574719515, 3124648750706, 13493792787415, 58422790497226, 253547380435914, 1102776319943605

Offset: 0

N. J. A. Sloane, Sep 14 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.

Cf. A128720 (2-ary case), A246974 (3-ary case).

a(n)={my(m=4); if(n<1, n==0, sum(k=1, n+1, sum(i=1, k-1, sum(j=0, (n-i)\m, (-1)^j*binomial(k, i)*binomial(i, j)*binomial(n-i, k-i-1)*binomial(n-m*j-1, i-1)))/k))} \\ Andrew Howroyd, Feb 24 2020

a(n) = Sum_{k=1..n+1} Sum_{i=1..k-1} Sum_{j=0..floor((n-i)/4)} (-1)^j*binomial(k, i)*binomial(i, j)*binomial(n-i, k-i-1)*binomial(n-4*j-1, i-1)/k for n > 0. - Andrew Howroyd, Feb 24 2020

Terms a(11) and beyond from Andrew Howroyd, Feb 24 2020

cp's OEIS Frontend

A059345 Central column of Pascal's "rhombus" (actually a triangle) A059317.

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

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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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A059398 Row sums of triangle in A059397.

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

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A174168 A (1,3) Somos-4 sequence.

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