A132276 -id:A132276 - cp's OEIS frontend

A190155 Central coefficients of the Riordan matrix (g(x),xg(x)), where g(x) = (1-x-x^2-sqrt(1-2x-5x^2+2x^3+x^4))/(2*x^2) (A132276).

1, 2, 12, 64, 385, 2346, 14672, 92936, 595179, 3841970, 24959726, 162988464, 1068860884, 7034520304, 46437268905, 307351081056, 2038878634695, 13552394472612, 90242046694715, 601847594327000, 4019556724362165, 26879647264387170

Offset: 0

Emanuele Munarini, May 05 2011

G. C. Greubel, Table of n, a(n) for n = 0..750 (terms 0..100 from Vincenzo Librandi)

Cf. A132276.

Table[Sum[Binomial[n+2i,i](n+1)/(i+n+1)Sum[Binomial[2n-j,n+2i]Binomial[n-2i-j,j],{j,0,n-2i}],{i,0,n/2}],{n,0,21}]

makelist(sum(binomial(n+2*i,i)*(n+1)/(i+n+1)*sum(binomial(2*n-j,n+2*i)*binomial(n-2*i-j,j),j,0,n-2*i),i,0,n/2),n,0,21);

for(n=0,30, print1(sum(i=0,n/2, binomial(n+2*i,i)*((n+1)/(i+n+1)) *sum(j=0, n-2*i, binomial(2*n-j,n+2*i)*binomial(n-2*i-j,j))), ", ")) \\ G. C. Greubel, Dec 28 2017

a(n) = T(2*n,n) where T(n,k) = A132276(n,k).

a(n) = Sum_{i=0..(n/2)} ( binomial(n+2*i,i)*((n+1)/(i+n+1)) * Sum_{j=0..(n-2*i)} binomial(2*n-j,n+2*i)*binomial(n-2*i-j,j) ).

A128720 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).

Original entry on oeis.org

1, 1, 3, 6, 16, 40, 109, 297, 836, 2377, 6869, 20042, 59071, 175453, 524881, 1579752, 4780656, 14536878, 44394980, 136107872, 418757483, 1292505121, 4001039563, 12418772656, 38641790001, 120510911885, 376628460529, 1179376013552, 3699860515924, 11626784875214

Offset: 0

Emeric Deutsch, Mar 30 2007, revised Sep 03 2007

nonn

Points of two kinds are placed on a line: light points having weight 1 and heavy points having weight 2. Number of configurations of points of total weight n, with some of the light points being paired off by nonintersecting arcs.
Number of skew Dyck paths of semilength n having no UUU's. 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 a path is defined to be the number of its steps. Example: a(3)=6 because we have UDUDUD, UDUUDD, UDUUDL, UUDDUD, UUDUDD and UUDUDL. a(n)=A128719(n,0). a(n)=A059397(n,n). a(n)=A132276(n,0).
Hankel transform is the (1,3) Somos-4 sequence A174168. - Paul Barry, Mar 10 2010
First column of the Riordan matrix A132276. - Emanuele Munarini, May 05 2011

			a(3)=6 because we have hhh, hH, Hh, hUD, UhD and UDh.
G.f. = 1 + x + 3*x^2 + 6*x^3 + 16*x^4 + 40*x^5 + 109*x^6 + 297*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000 (first 101 terms from Vincenzo Librandi)
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, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Emeric Deutsch, Emanuele Munarini, and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
P. Rajkovic, Paul 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. A001006, A128719, A059397, A132276.

a[0]:=1: a[1]:=1: for n from 2 to 30 do a[n]:=a[n-1]+a[n-2]+add(a[j]*a[n-2-j], j=0..n-2) end do: seq(a[n],n=0..30); G:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/z^2: Gser:=series(G,z=0,33): seq(coeff(Gser,z,n),n=0..30);

Table[Sum[Binomial[2k,k]/(k+1)Sum[Binomial[n-j,2k]Binomial[n-j-2k,j],{j,0,n/2}],{k,0,n/2}],{n,0,12}] (* Emanuele Munarini, May 05 2011 *)

makelist(sum(binomial(2*k,k)/(k+1)*sum(binomial(n-j,2*k)*binomial(n-j-2*k,j),j,0,n/2),k,0,n/2),n,0,12); /* Emanuele Munarini, May 05 2011 */

a(n) = Sum_{j=0..floor(n/2)} binomial(n-j, j)*m(n-2j), where m(k)=A001006(k) are the Motzkin numbers.
G.f. = G satisfies z^2*G^2 - (1-z-z^2)*G + 1 = 0.
G.f. = c(z^2/(1-z-z^2)^2)/(1-z-z^2), where c(z) = (1-sqrt(1-4z))/(2z) is the Catalan function.
a(n) = a(n-1) + a(n-2) + Sum_{j=0..n-2} a(j)*a(n-2-j), a(0) = a(1) = 1.
G.f.: (1/(1-x-x^2))*c(x^2/(1-x-x^2)^2) = (1/(1-x^2))*m(x/(1-x^2)), c(x) the g.f. of A000108, m(x) the g.f. of A001006. - Paul Barry, Mar 18 2010
Let A(x) be the g.f., then B(x) = 1 + x*A(x) = 1 + 1*x + 1*x^2 + 3*x^3 + 6*x^4 + ... = 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1+x-x^2) (continued fraction); more generally B(x)=C(x/(1+x-x^2)) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011
a(n) = Sum_{k=0..floor(n/2)} (binomial(2*k,k)/(k+1))*Sum_{j=0..floor(n/2)} binomial(n-j, 2*k)*binomial(n-j-2*k, j). - Emanuele Munarini, May 05 2011
D-finite with recurrence: (n+2)*a(n) + (-2*n-1)*a(n-1) + 5*(-n+1)*a(n-2) + (2*n-5)*a(n-3) + (n-4)*a(n-4) = 0. - R. J. Mathar, Dec 03 2012
G.f.: (1 - x - x^2 - sqrt(1 - 2*x - 5*x^2 + 2*x^3 + x^4))/(2*x^2) = 1/Q(0), where Q(k) = 1 - x - x^2 - x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2013
a(n) ~ sqrt(78+22*sqrt(13)) * ((3+sqrt(13))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014

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

A190156 Expansion of (1-x-3x^2-sqrt(1-2x-5x^2+2x^3+x^4))/(2x^3(1+2*x)).

Original entry on oeis.org

1, 1, 4, 8, 24, 61, 175, 486, 1405, 4059, 11924, 35223, 105007, 314867, 950018, 2880620, 8775638, 26843704, 82420464, 253916555, 784672011, 2431695541, 7555381574, 23531026853, 73448858179, 229730744171, 719914525210, 2260031465504, 7106721944206

Offset: 0

Emanuele Munarini, May 05 2011

nonn

Diagonal 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).

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.

Cf. A132276.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x-3*x^2-Sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)))); // G. C. Greubel, Oct 22 2018

T := (n, k) -> simplify(2^k*binomial(n-k, k)*hypergeom([-k, k-n-1], [2], 1/2)):
seq(add(T(n, k), k=0..floor(n/2)), n=0..28); # Peter Luschny, Oct 19 2020

CoefficientList[Series[(1-x-3x^2-Sqrt[1-2x-5x^2+2x^3+x^4])/(2x^3(1+2x)),{x,0,28}],x]

a(n):=sum(sum(2^(m-j+1)*binomial(n-m,j-1)*binomial(n-m+2,j)*binomial(n-m-j+1,m-j+1),j,0,n-m+2)/(n-m+2),m,0,n); /* Vladimir Kruchinin, Oct 19 2020 */

x='x+O('x^66); Vec((1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x))) /* Joerg Arndt, May 15 2011 */

G.f.: (1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)).
D-finite with recurrence: (n+3)*a(n) +3*a(n-1) +3*(-3*n-2)*a(n-2) +(-8*n-3)*a(n-3) +(5*n-9) *a(n-4) +2*(n-3)*a(n-5)=0. - R. J. Mathar, Oct 08 2016
a(n) = Sum_{m=0..n} Sum_{j=0..n-m+2} 2^(m-j+1)*C(n-m,j-1)*C(n-m+2,j)*C(n-m-j+1,m-j+1)/(n-m+2). - Vladimir Kruchinin, Oct 19 2020
a(n) = Sum_{k=0..floor(n/2)} 2^k*binomial(n-k, k)*hypergeom([-k, k-n-1], [2], 1/2). - Peter Luschny, Oct 19 2020

cp's OEIS Frontend

A190155 Central coefficients 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).

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A128720 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).

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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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A190156 Expansion of (1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)).

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A190155 Central coefficients of the Riordan matrix (g(x),xg(x)), where g(x) = (1-x-x^2-sqrt(1-2x-5x^2+2x^3+x^4))/(2*x^2) (A132276).

A190156 Expansion of (1-x-3x^2-sqrt(1-2x-5x^2+2x^3+x^4))/(2x^3(1+2*x)).