A060693 -id:A060693 - cp's OEIS frontend

A143330 G.f. satisfies: A(x) = (1 + x*A(x)^2)/(1 - x^2).

1, 1, 3, 8, 25, 83, 289, 1041, 3847, 14504, 55569, 215727, 846761, 3354858, 13398965, 53888063, 218053915, 887107888, 3626373205, 14887942624, 61358959587, 253771944529, 1052917272543, 4381374717994, 18280470530047, 76459765772375

Offset: 0

Paul D. Hanna, Aug 08 2008

Diagonal sums of A060693. - Paul Barry, Feb 11 2009
Starting with the second 1 and inserting a 2 between the 1 and 3: (1, 2, 3, 8, 25, 83, ...) the INVERT transform of that sequence deletes the 2, getting (1, 3, 8, 25, 83, ...). - Gary W. Adamson, Jun 24 2015
Number of Schroeder-like (see A006318) excursions (paths on or above height 0 beginning and ending at height 0) of semilength n, with steps U=(1,1), D=(1,-1), and H=(4,0). - Alexander Burstein, May 21 2025
a(n) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, all internal nodes have weight 1, and leaf nodes have weights in {1,2}. - John Tyler Rascoe, Jun 06 2025

			G.f. = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 + 1041*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Joseph P. S. Kung and Anna de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From _N. J. A. Sloane_, Dec 27 2012

Cf. A000108, A006318, A060693, A073157.

a := proc(n) option remember; if n <= 3 then return [1, 1, 3, 8][n + 1] fi;
((5 - n)*a(n - 4) + (2*n - 4)*a(n - 2) + (4*n - 2)*a(n - 1))/(n + 1) end:
seq(a(n), n = 0..25); # Peter Luschny, Jan 25 2023

CoefficientList[Series[(1 - x^2 - Sqrt[1 - 4 x - 2 x^2 + x^4])/(2 x), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=polcoeff((1-x^2-sqrt((1-x^2)^2-4*x+x^2*O(x^n)))/(2*x),n)}

G.f.: A(x) = (1-x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x).
From Paul Barry, Feb 11 2009: (Start)
G.f.: 1/(1-x^2-x/(1-x^2-x/(1-x^2-x/(1-x^2-x/(1-...))))) (continued fraction).
a(n) = Sum_{k=0..floor(n/2)} C(2n-3k,k)*A000108(n-2k). (End)
D-finite with recurrence (n+1)*a(n) +(n+2)*a(n-1) +2*(17-11n)*a(n-2) +10*(3-n)*a(n-3) +(n-5)*a(n-4) +5*(n-6)*a(n-5)=0. - R. J. Mathar, Dec 11 2011
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 4.439109106851354261627... is the root of the equation 1 - 2*d^2 - 4*d^3 + d^4 = 0 and c = 1/2*sqrt(d*(d^2+3)/(d^2-1)) = 1.16064231... - Vaclav Kotesovec, Feb 03 2014
G.f. satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} binomial(j,k)*x^k*A(x)^(j-k). - Ilya Gutkovskiy, Apr 11 2019
G.f.: 1/G(x), with G(x) = 1-(x+x^2)/(1-x/G(x)) (continued fraction). - Nikolaos Pantelidis, Jan 11 2023
From Peter Luschny, Jan 25 2023: (Start)
a(n) = CatalanNumber(n)*hypergeom([-n/2, -n/2, -n/2 - 1/2, -n/2 + 1/2], [-(2*n)/3, -(2*n)/3 + 1/3, -(2*n)/3 + 2/3], -16/27).
a(n) = ((5 - n)*a(n - 4) + (2*n - 4)*a(n - 2) + (4*n - 2)*a(n - 1))/(n + 1) for n >= 4. (End)
G.f. A(x) = -x + (1/x)*series_reversion(x*G(-x)), where G(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 + ... is the g.f. of A073157. - Peter Bala, Aug 27 2024

Minor edits by Vaclav Kotesovec, Mar 31 2014

A133305 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)4^i5^(n-i), a(0) = 1.

Original entry on oeis.org

1, 5, 45, 505, 6345, 85405, 1204245, 17558705, 262577745, 4005148405, 62070886845, 974612606505, 15471084667545, 247876665109005, 4003225107031845, 65101209768055905, 1065128963164067745, 17520376884067071205, 289572455530026439245, 4806489064223483202905

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Fifth column of array A103209.

The Hankel transform of this sequence is 20^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..795
Samuele Giraudo, Operads from posets and Koszul duality, arXiv:1504.04529 [math.CO], 2015-2016.

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-18*x+1))/(8*x))) // G. C. Greubel, Feb 10 2018

a[n_] := Hypergeometric2F1[-n, n + 1, 2, -4];
Table[a[n], {n, 0, 16}] (* Peter Luschny, Jan 08 2018 *)
CoefficientList[Series[(1-x-Sqrt[x^2-18*x+1])/(8*x), {x, 0, 50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-18*x+1))/(8*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-18*z+1))/(8*z).
a(n) = Sum_{k=0..n} A088617(n,k)*4^k.
a(n) = Sum_{k=0..n} A060693(n,k)*4^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)*4^k*C(k), C(n) given by A000108.
a(0) = 1, a(n) = a(n-1) + 4*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 9*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
G.f.: 1/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) = hypergeom([-n, n + 1], [2], -4). - Peter Luschny, Jan 08 2018
a(n) ~ 5^(1/4) * phi^(6*n + 3) / (2^(5/2) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 21 2021

A154825 Reversion of x(1-2x)/(1-3*x).

Original entry on oeis.org

1, -1, -1, 1, 5, 3, -21, -51, 41, 391, 407, -1927, -6227, 2507, 49347, 71109, -236079, -966129, 9519, 7408497, 13685205, -32079981, -167077221, -60639939, 1209248505, 2761755543, -4457338681, -30629783831, -22124857219, 206064020315, 572040039283, -590258340811

Offset: 0

Paul Barry, Jan 15 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A060693, A086810, A088617, A090181, A091593, A131198, A133336.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1+3*x-Sqrt(1-2*x+9*x^2))/(4*x) )); // G. C. Greubel, May 24 2022

A154825_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := -a[w-1]+2*add(a[j]*a[w-j-1],j=1..w-1)od;
convert(a, list) end: A154825_list(28); # Peter Luschny, May 19 2011

CoefficientList[Series[(1+3*x-Sqrt[1-2*x+9*x^2])/(4*x), {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

[sum(binomial(n+k,n-k)*catalan_number(k)*2^k*(-3)^(n-k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, May 24 2022

G.f.: (1+3*x-sqrt(1-2*x+9*x^2))/(4*x). - corrected by Vaclav Kotesovec, Feb 08 2014
G.f.: 1/(1+x/(1-2x/(1+x/(1-2x/(1+x/(1-2x/(1+.... (continued fraction).
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*A000108(k)*2^k*(-3)^(n-k).
From Philippe Deléham, Jan 17 2009: (Start)
a(n) = Sum_{k=0..n} A131198(n,k)*(-1)^(n-k)*2^k.
a(n) = Sum_{k=0..n} A090181(n,k)*(-1)^k*2^(n-k).
a(n) = Sum_{k=0..n} A060693(n,k)*2^(n-k)*(-3)^k.
a(n) = Sum_{k=0..n} A088617(n,k)*2^k*(-3)^(n-k).
a(n) = Sum_{k=0..n} A086810(n,k)*(-1)^k*3^(n-k).
a(n) = Sum_{k=0..n} A133336(n,k)*3^k*(-1)^(n-k). (End)
D-finite with recurrence (n+1)*a(n) = (2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - R. J. Mathar, Nov 15 2012
a(n) = (-3)^n*Hypergeometric2F1([-n, n+1], [2]; 2/3). - G. C. Greubel, May 24 2022

A133306 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)5^i6^(n-i), a(0)=1.

Original entry on oeis.org

1, 6, 66, 906, 13926, 229326, 3956106, 70572066, 1291183806, 24095736726, 456879955026, 8776867331706, 170459895028566, 3341423256586206, 66023812564384026, 1313634856606430226, 26295597219228901806, 529199848207277494566, 10701116421278640683106, 217317899302044152030826

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Sixth column of array A103209.

The Hankel transform of this sequence is 30^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..745

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-22*x+1))/(10*x))) // G. C. Greubel, Feb 10 2018

CoefficientList[Series[(1-x-Sqrt[x^2-22*x+1])/(10*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-22*x+1))/(10*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-22*z+1))/(10*z).
a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*5^k.
a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*5^(n-k).
a(n) = Sum_{k, 0<=k<=n} C(n+k, 2*k) 5^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 5*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 11*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
G.f.: 1/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) ~ 3^(1/4) * (11 + 2*sqrt(30))^(n + 1/2) / (10^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 29 2021

A133307 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)6^i7^(n-i), a(0)=1.

Original entry on oeis.org

1, 7, 91, 1477, 26845, 522739, 10663471, 224939113, 4866571801, 107393779423, 2407939176643, 54700070934061, 1256249370578293, 29119953189833611, 680401905145643863, 16008309928027493713, 378930780842531820721, 9017843351806985482423, 215634517504141993966891

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Seventh column of array A103209.

The Hankel transform of this sequence is 42^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..700

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-26*x+1))/(12*x))) // G. C. Greubel, Feb 10 2018

a := n -> hypergeom([-n, n+1], [2], -6);
seq(round(evalf(a(n),32)),n=0..16); # Peter Luschny, May 23 2014

CoefficientList[Series[(1-x-Sqrt[x^2-26*x+1])/(12*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-26*x+1))/(12*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-26*z+1))/(12*z).
a(n) = Sum_{k=0..n} A088617(n,k)*6^k .
a(n) = Sum_{k=0..n} A060693(n,k)*6^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)6^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 6*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 13*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
a(n) = hypergeom([-n, n+1], [2], -6). # Peter Luschny, May 23 2014
G.f.: 1/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) ~ 42^(1/4) * (13 + 2*sqrt(42))^(n + 1/2) / (12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 29 2021

A133308 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)7^i8^(n-i), a(0)=1.

Original entry on oeis.org

1, 8, 120, 2248, 47160, 1059976, 24958200, 607693640, 15175702200, 386555020552, 10004252294520, 262321706465736, 6953918939056440, 186059575955360136, 5018045415643478520, 136276936332343342152, 3723442515218861494200, 102281105054908404972040

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Eighth column of array A103209.

The Hankel transform of this sequence is 56^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..675

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-30*x+1))/(14*x))) // G. C. Greubel, Feb 10 2018

a := n -> hypergeom([-n, n+1], [2], -7);
seq(round(evalf(a(n), 32)), n=0..15); # Peter Luschny, May 23 2014

CoefficientList[Series[(1-x-Sqrt[x^2-30*x+1])/(14*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-30*x+1))/(14*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-30*z+1))/(14*z).
a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*7^k.
a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*7^(n-k).
a(n) = Sum_{k, 0<=k<=n} C(n+k, 2k)7^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 7*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 15*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
a(n) = hypergeom([-n, n+1], [2], -7). - Peter Luschny, May 23 2014
G.f.: 1/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

A133309 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)8^i9^(n-i), a(0)=1.

Original entry on oeis.org

1, 9, 153, 3249, 77265, 1968633, 52546473, 1450365921, 41058670113, 1185580310121, 34783088255289, 1033907690362257, 31070005849929969, 942384250116160857, 28812102048874578249, 887007207177728561601, 27473495809057571051073, 855518113376312857290441

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Ninth column of array A103209.

The Hankel transform of this sequence is 72^C(n+1,2). - Philippe Deléham, Oct 29 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!( (1-x-Sqrt(x^2-34*x+1))/16 )); // G. C. Greubel, Feb 10 2018

Rest@ CoefficientList[ Series[(1-x-Sqrt[x^2-34*x+1])/16, {x, 0, 18}], x] (* Robert G. Wilson v, Oct 19 2007 *)
Table[-((3 I LegendreP[n, -1, 2, 17])/(2 Sqrt[2])), {n, 0, 20}] (* Vaclav Kotesovec, Aug 13 2013 *)

my(x='x+O('x^30)); Vec((1-x-sqrt(x^2-34*x+1))/16) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-34*z+1))/16.
a(n) = Sum_{k=0..n} A088617(n,k)*8^k.
a(n) = Sum_{k=0..n} A060693(n,k)*8^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)8^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 8*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
a(n) ~ sqrt(144+102*sqrt(2))*(17+12*sqrt(2))^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013
Recurrence: (n+1)*a(n) = 17*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Aug 13 2013
G.f.: 1/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

More terms from Robert G. Wilson v, Oct 19 2007

A152601 a(n) = Sum_{k=0..n} C(n+k,2k)A000108(k)3^k*2^(n-k).

Original entry on oeis.org

1, 5, 40, 395, 4360, 51530, 637840, 8163095, 107140360, 1434252230, 19507077040, 268796321870, 3744480010960, 52647783144980, 746145741252640, 10648007952942095, 152877753577617160, 2206713692628578030

Offset: 0

Paul Barry, Dec 09 2008

Hankel transform is 15^C(n+1,2).

Cf. A103211, A103210.
Cf. A088617, A060693, A090181, A131198, A133336, A086810
Cf. A152600.

a(n) = A152600(n+1)/2.
a(n) = Sum_{k=0..n} A088617(n,k)*3^k*2^(n-k) = Sum_{k=0..n} A060693(n,k)*2^k*3^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A090181(n,k)*5^k*3^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A131198(n,k)*3^k*5^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A133336(n,k)*(-2)^k*5^(n-k) = Sum_{k=0..n} A086810(n,k)*5^k*(-2)^(n-k). - Philippe Deléham, Dec 10 2008
G.f.: 1/(1-5x/(1-3x/(1-5x/(1-3x/(1-5x/(1-3x/(1-5x/(1-... (continued fraction). - Philippe Deléham, Nov 28 2011
Conjecture: (n+1)*a(n) +8*(-2*n+1)*a(n-1) +4*(n-2)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
G.f.: 1/G(x), with G(x) = 1-2*x-(3*x)/G(x) (continued fraction). - Nikolaos Pantelidis, Jan 09 2023

A101282 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k valleys.

Original entry on oeis.org

2, 5, 1, 14, 7, 1, 42, 36, 11, 1, 132, 165, 80, 16, 1, 429, 715, 484, 155, 22, 1, 1430, 3003, 2639, 1183, 273, 29, 1, 4862, 12376, 13468, 7840, 2554, 448, 37, 1, 16796, 50388, 65688, 47328, 20124, 5031, 696, 46, 1, 58786, 203490, 310080, 267444, 141219, 46377, 9230, 1035, 56, 1

Offset: 1

Emeric Deutsch, Dec 20 2004

A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis (Schroeder paths are counted by the large Schroeder numbers (A006318)). Also number of Schroeder paths of length 2n and having k UU's. Also number of Schroeder paths of length 2n and having k peaks at height >1,

			T(3,1) = 7 because we have HU(DU)D, U(DU)DH, U(DU)HD, UH(DU)D, U(DU)UDD, UUD(DU)D and UU(DU)DD, the valleys being shown between parentheses.
Triangle begins:
    2;
    5,   1;
   14,   7,  1;
   42,  36, 11,  1;
  132, 165, 80, 16, 1;
  ...

Alois P. Heinz, Rows n = 1..150, flattened
Wikipedia, Schröder number

Row sums give A006318.
T(2n,n) gives A385299.
Cf. A000108, A003516, A060693.

G := 1/2/(-t*z-z^2+z^2*t)*(-1+2*z-t*z+sqrt(1-4*z-2*t*z+t^2*z^2)):Gser:=simplify(series(G,z=0,13)):for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 11 do seq(coeff(t*P[n],t^k),k=1..n) od; # yields the sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
     `if`(x=0, 1, b(x-1, y-1, 1)+b(x-1, y+1, 0)*z^t+b(x-2, y, 0))))
    end:
T:= (n, k)-> coeff(b(2*n, 0$2), z, k):
seq(seq(T(n,k), k=0..n-1), n=1..12);  # Alois P. Heinz, Jun 17 2025

T(n,m):=if n=0 or m=0 then 0 else if m=1 then 1/(n+1)*binomial(2*n+2,n) else  sum(((k+1)*binomial(n-k,m-1)*binomial(2*n-m-k+1,n+1))/(n-k),k,0,n-m); /* Vladimir Kruchinin, Oct 14 2020 */

G.f.: G=G(t, z) satisfies z(t+z-tz)G^2-(1-2z+tz)G+1=0.
T(n,m) = Sum_{k=0..n-m} (k+1)*C(n-k,m-1)*C(2*n-m-k+1,n+1)/(n-k), m>1,  T(n,1) = 1/(n+1)*binomial(2*n+2,n).  - Vladimir Kruchinin, Oct 14 2020
From Mikhail Kurkov, Jun 17 2025: (Start)
Conjecture: The n-th row polynomial is R(n+1,0) where
  R(n,n) = 1,
  R(n,0) = Sum_{j=0..n-1} R(n-1,j) for n > 0,
  R(n,k) = R(n-1,k-1) + (x+1) * (R(n,0) - Sum_{j=0..k-1} R(n-1,j)) for 0 < k < n. (End)

A190907 Triangle read by rows: T(n,k) = binomial(n+k, n-k) k! / (floor(k/2)! * floor((k+2)/2)!).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 3, 1, 10, 15, 21, 2, 1, 15, 35, 84, 18, 10, 1, 21, 70, 252, 90, 110, 5, 1, 28, 126, 630, 330, 660, 65, 35, 1, 36, 210, 1386, 990, 2860, 455, 525, 14, 1, 45, 330, 2772, 2574, 10010, 2275, 4200, 238, 126

Offset: 0

Peter Luschny, May 24 2011

The triangle may be regarded as a generalization of the triangle A088617.
A088617(n,k) = binomial(n+k,n-k)*(2*k)$/(k+1);
T(n,k) = binomial(n+k,n-k)*(k)$ /(floor(k/2)+1).
Here n$ denotes the swinging factorial A056040(n). As A088617 is a decomposition of the large Schroeder numbers A006318, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.
T(n,n) = A057977(n) which can be seen as extended Catalan numbers.

			[0]  1
[1]  1,  1
[2]  1,  3,   1
[3]  1,  6,   5,   3
[4]  1, 10,  15,  21,   2
[5]  1, 15,  35,  84,  18,  10
[6]  1, 21,  70, 252,  90, 110,  5
[7]  1, 28, 126, 630, 330, 660, 65, 35

Peter Luschny, The lost Catalan numbers.

Cf. Row sums: A190908; A056040, A085478, A088617, A060693.

A190907 := (n,k) -> binomial(n+k,n-k)*k!/(floor(k/2)!*floor((k+2)/2)!);
seq(print(seq(A190907(n,k), k=0..n)), n=0..7);

Flatten[Table[Binomial[n+k,n-k] k!/(Floor[k/2]!Floor[(k+2)/2]!),{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 05 2012 *)

T(n,1) = A000217(n). T(n,2) = (n-1)*n*(n+1)*(n+2)/24 (Cf. A000332).

cp's OEIS Frontend

A143330 G.f. satisfies: A(x) = (1 + x*A(x)^2)/(1 - x^2).

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A133305 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*4^i*5^(n-i), a(0) = 1.

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Magma

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A154825 Reversion of x*(1-2*x)/(1-3*x).

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A133306 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*5^i*6^(n-i), a(0)=1.

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A133307 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*6^i*7^(n-i), a(0)=1.

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A133308 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*7^i*8^(n-i), a(0)=1.

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A133309 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*8^i*9^(n-i), a(0)=1.

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A133305 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)4^i5^(n-i), a(0) = 1.

A154825 Reversion of x(1-2x)/(1-3*x).

A133306 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)5^i6^(n-i), a(0)=1.

A133307 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)6^i7^(n-i), a(0)=1.

A133308 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)7^i8^(n-i), a(0)=1.

A133309 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)8^i9^(n-i), a(0)=1.

A152601 a(n) = Sum_{k=0..n} C(n+k,2k)A000108(k)3^k*2^(n-k).