A088617 -id:A088617 - cp's OEIS frontend

A123160 Triangle read by rows: T(n,k) = n!(n+k-1)!/((n-k)!(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.

1, 1, 1, 1, 4, 3, 1, 9, 18, 10, 1, 16, 60, 80, 35, 1, 25, 150, 350, 350, 126, 1, 36, 315, 1120, 1890, 1512, 462, 1, 49, 588, 2940, 7350, 9702, 6468, 1716, 1, 64, 1008, 6720, 23100, 44352, 48048, 27456, 6435, 1, 81, 1620, 13860, 62370, 162162, 252252, 231660, 115830, 24310

Offset: 0

Roger L. Bagula, Oct 02 2006

T(n,k) is also the number of order-preserving partial transformations (of an n-element chain) of width k (width(alpha) = |Dom(alpha)|). - Abdullahi Umar, Aug 25 2008

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  9,  18,  10;
  1, 16,  60,  80,  35;
  1, 25, 150, 350, 350, 126;
  ...

Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965 pages 296 and 305

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.

Cf. A005773, A037965, A059481, A085373, A088218, A088617, A122899, A123164, A178301.

[Binomial(n,k)*Binomial(n+k-1,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2022

T:=proc(n,k) if k=0 and n=0 then 1 elif k<=n then n!*(n+k-1)!/(n-k)!/(n-1)!/(k!)^2 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[n_, m_]= If [n==m==0, 1, n!*(n+m-1)!/((n-m)!*(n-1)!(m!)^2)];
Table[T[n, m], {n,0,10}, {m,0,n}]//Flatten
max = 9; s = (x+1)/(2*Sqrt[(1-x)^2-4*y])+1/2 + O[x]^(max+2) + O[y]^(max+2); T[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]; Table[T[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 18 2015, after Vladimir Kruchinin *)

def A123160(n,k): return binomial(n, k)*binomial(n+k-1, k)
flatten([[A123160(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2022

T(n, m) = n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2), with T(0, 0) = 1.
T(n, k) = binomial(n,k)*binomial(n+k-1,k). The row polynomials (except the first) are (1+x)*P(n,0,1,2x+1), where P(n,a,b,x) denotes the Jacobi polynomial. The columns of this triangle give the diagonals of A122899. - Peter Bala, Jan 24 2008
T(n, k) = binomial(n,k)*(n+k-1)!/((n-1)!*k!).
T(n, k)= binomial(n,k)*binomial(n+k-1,n-1). - Abdullahi Umar, Aug 25 2008
G.f.: (x+1)/(2*sqrt((1-x)^2-4*y)) + 1/2. - Vladimir Kruchinin, Jun 16 2015
From _Peter Bala, Jul 20 2015: (Start)
O.g.f. (1 + x)/( 2*sqrt((1 - x)^2 - 4*x*y) ) + 1/2 = 1 + (1 + y)*x + (1 + 4*y + 3*y^2)*x^2 + ....
For n >= 1, the n-th row polynomial R(n,y) = (1 + y)*r(n-1,y), where r(n,y) is the n-th row polynomial of A178301.
exp( Sum_{n >= 1} R(n,y)*x^n/n ) = 1 + (1 + y)*x + (1 + 3*y + 2*y^2)*x^2 + ... is the o.g.f for A088617. (End)
From G. C. Greubel, Jun 19 2022: (Start)
T(n, n) = A088218(n).
T(n, n-1) = A037965(n).
T(n, n-2) = A085373(n-2).
Sum_{k=0..n} T(n, k) = A123164(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A005773(n). (End)

Edited by N. J. A. Sloane, Oct 26 2006 and Jul 03 2008

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

A088626 a(n) = 42*binomial(n,10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 462, 2772, 12012, 42042, 126126, 336336, 816816, 1837836, 3879876, 7759752, 14814072, 27159132, 48050772, 82372752, 137287920, 223092870, 354323970, 551170620, 841260420, 1261890630, 1862790930, 2709514080, 3887563680

Offset: 0

N. J. A. Sloane, Nov 23 2003

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

A diagonal of A088617.

42Binomial[Range[0,40],10] (* Harvey P. Dale, Apr 19 2020 *)

a(n)=42*binomial(n,10) \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 42*x^10/(1-x)^11. - Colin Barker, Oct 30 2012

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

A140662 Number of possible column states for self-avoiding polygons in a slit of width n.

Original entry on oeis.org

1, 3, 8, 20, 50, 126, 322, 834, 2187, 5797, 15510, 41834, 113633, 310571, 853466, 2356778, 6536381, 18199283, 50852018, 142547558, 400763222, 1129760414, 3192727796, 9043402500, 25669818475, 73007772801, 208023278208, 593742784828, 1697385471210, 4859761676390

Offset: 1

R. J. Mathar, Jul 11 2008

Number of Dyck (n+1)-paths whose maximum ascent length is 2. - David Scambler, Aug 22 2012

Number of (n+1)-Motzkin-paths with at least one up-step (see A001006 and the Python program). - Peter Luschny, Dec 03 2024

			The 20 Motzkin-paths of length 5 with at least one up-step are: UUDDF, UUDFD, UUFDD, UDUDF, UDUFD, UDFUD, UDFFF, UFUDD, UFDUD, UFDFF, UFFDF, UFFFD, FUUDD, FUDUD, FUDFF, FUFDF, FUFFD, FFUDF, FFUFD, FFFUD.

J. Alvarez, E.J. Janse van Rensburg et al. Self-avoiding walks and polygons in slits.
Xiaomei Chen, Counting humps and peaks in Motzkin paths with height k, arXiv:2412.00668 [math.CO], 2024. See p. 7.
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 148.

Cf. A055151, A080159, A088617, A107131, A097610.
Cf. A001006.
Column k=2 of A203717 (shifted).

a := n -> n*(n + 1)*hypergeom([1, -n/2 + 1, 1/2 - n/2], [2, 3], 4)/2:
seq(simplify(a(n)), n = 1..30);  # Peter Luschny, Dec 03 2024

# A generator of the Motzkin-paths with at least one up-step.
C = str.count
def aGen(n: int): # -> Generator[str, Any, list[str]]
    a = [""]
    for w in a:
        if len(w) == n + 1:
            if (C(w, "U") > 0 and C(w, "U") == C(w, "D")): yield w
        else:
            for j in "UDF":
                u = w + j
                if C(u, "U") >= C(u, "D"): a += [u]
    return a
for n in range(1, 6):
    SAP = [w for w in aGen(n)]
    print(len(SAP), ":", SAP)  # Peter Luschny, Dec 03 2024

a(n) = Sum_{m=1..[(n+1)/2]} (n+1)!/((n+1-2m)!m!(m+1)!).
a(n) = A001006(n + 1) - 1. [Corrected by Peter Luschny, Dec 03 2024]
D-finite with recurrence (n+3)*a(n) + (-4*n-7)*a(n-1) + (2*n+3)*a(n-2) + (4*n-5)*a(n-3) + 3*(-n+2)*a(n-4) = 0. - R. J. Mathar, Nov 01 2021
From Peter Luschny, Dec 03 2024: (Start)
a(n) = (1/2)*n*(n + 1)*hypergeom([1, -n/2 + 1, 1/2 - n/2], [2, 3], 4).
a(n) = n!*[x^n]((exp(x)*(-x^3 + 2*(2*x - 3)*x*BesselI(0,2*x) + (x*(5*x - 4) + 6)*BesselI(1, 2* x)))/x^3). (End)

A088625 14*C(n,8).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 14, 126, 630, 2310, 6930, 18018, 42042, 90090, 180180, 340340, 612612, 1058148, 1763580, 2848860, 4476780, 6864396, 10296594, 15142050, 21871850, 31081050, 43513470, 60090030, 81940950, 110442150, 147256200, 194378184, 254186856, 329501480

Offset: 0

N. J. A. Sloane

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

A diagonal of A088617.

G.f.: 14*x^8 / (1-x)^9. [Colin Barker, Dec 19 2012]

Revised Nov 23 2003, Jun 12 2012

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

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A123160 Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.

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

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A088626 a(n) = 42*binomial(n,10).

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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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A123160 Triangle read by rows: T(n,k) = n!(n+k-1)!/((n-k)!(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 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).