A106566 -id:A106566 - cp's OEIS frontend

A127628 G.f. 1/(1-6xc(x)) where c(x) is the g.f. of A000108.

1, 6, 42, 300, 2154, 15492, 111492, 802584, 5778090, 41600532, 299517996, 2156509416, 15526797252, 111792690600, 804906480840, 5795323452720, 41726317225770, 300429441596340, 2163091823919900, 15574260559056840, 112134673904493420, 807369644235408120

Offset: 0

Paul Barry, Jan 20 2007

nonn

Image of 6^n under the Catalan transform g(x)->g(xc(x)). The Hankel transform of this sequence and of the aerated version with g.f. 1/(1-6*x^2*c(x^2)) is 6^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n.

Robert Israel, Table of n, a(n) for n = 0..1165

Cf. A000108, A039599, A106566.

f:= gfun:-rectoproc({(72 + 144*n)*a(n) + (-82 - 56*n)*a(n + 1) + (5*n + 10)*a(n + 2), a(0) = 1, a(1) = 6},a(n),remember):
map(f, [$0..50]); # Robert Israel, Aug 28 2020

RecurrenceTable[{(-56*n - 82)*a[n + 1] + (5*n +  10)*a[n + 2] + (144*n + 72)*a[n] == 0, a[0] == 1, a[1] == 6}, a, {n, 0, 50}] (* Jean-François Alcover, Sep 15 2022, after Robert Israel *)

my(x='x+O('x^25)); Vec(1/(1-6*x*(1-sqrt(1-4*x))/(2*x))) \\ Michel Marcus, Sep 15 2022

a(n) = 1 if n=0, Sum_{k=1..n} C(2*n-k-1,n-k)*k*6^k/n otherwise;
a(n) = Sum_{k=0..n} C(2*n,n-k)*(2*k+1)*5^k/(n+k+1).
a(n) = Sum_{k=0..n} A106566(n,k)*6^k. - Philippe Deléham, Feb 04 2007
a(n) = Sum_{k=0..n} A039599(n,k)*5^k. - Philippe Deléham, Sep 08 2007
a(n) = (36*a(n-1) - 6*A000108(n-1))/5 for n >= 1, a(0) = 1. - Philippe Deléham, Nov 27 2007
Conjecture: 5*n*a(n) + 2*(15-28*n)*a(n-1) + 72*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
G.f.: (2+3*sqrt(1-4*x))/(5-36*x).  Mathar's conjecture verified using the differential equation (144*x^2-56*x+5)*y'+(72*x-26)*y = 4 satisfied by the g.f. - Robert Israel, Aug 28 2020

A110510 Riordan array (1, x*c(2x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 8, 4, 1, 0, 40, 20, 6, 1, 0, 224, 112, 36, 8, 1, 0, 1344, 672, 224, 56, 10, 1, 0, 8448, 4224, 1440, 384, 80, 12, 1, 0, 54912, 27456, 9504, 2640, 600, 108, 14, 1, 0, 366080, 183040, 64064, 18304, 4400, 880, 140, 16, 1, 0, 2489344, 1244672, 439296

Offset: 0

Paul Barry, Jul 24 2005

Row sums are C(2;n), A064062. Inverse is A110509. Diagonal sums are A108308. [Corrected by Philippe Deléham, Nov 09 2007]

Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 2, 2, 2, 2, 2, 2, 2, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 23 2014

			Rows begin
  1;
  0,   1;
  0,   2,   1;
  0,   8,   4,   1;
  0,  40,  20,   6,   1;
  0, 224, 112,  36,   8,   1;
  ...
Production matrix begins:
  0,  1;
  0,  2,  1;
  0,  4,  2,  1;
  0,  8,  4,  2,  1;
  0, 16,  8,  4,  2,  1;
  0, 32, 16,  8,  4,  2,  1;
  0, 64, 32, 16,  8,  4,  2,  1;
  ... - _Philippe Deléham_, Sep 23 2014

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[n_, k_] := (k/n)*Binomial[2*n - k - 1, n - k]*2^(n - k); Join[{1}, Table[T[n, k], {n, 1, 10}, {k, 0, n}]] // Flatten (* G. C. Greubel, Aug 29 2017 *)

concat([1], for(n=1,25, for(k=0,n, print1((k/n)*binomial(2*n-k-1, n-k)*2^(n-k), ", ")))) \\ G. C. Greubel, Aug 29 2017

Number triangle: T(0,k) = 0^k, T(n,k) = (k/n)*C(2n-k-1, n-k)*2^(n-k), n, k > 0.
T(n,k) = A106566(n,k)*2^(n-k). - Philippe Deléham, Nov 08 2007
T(n,k) = 2*T(n,k+1) + T(n-1,k-1) with T(n,n) = 1 and T(n,0) = 0 for n >= 1. - Peter Bala, Feb 02 2020

A110518 Riordan array (1, x*c(3x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 18, 6, 1, 0, 135, 45, 9, 1, 0, 1134, 378, 81, 12, 1, 0, 10206, 3402, 756, 126, 15, 1, 0, 96228, 32076, 7290, 1296, 180, 18, 1, 0, 938223, 312741, 72171, 13365, 2025, 243, 21, 1, 0, 9382230, 3127410, 729729, 138996, 22275, 2970, 315, 24, 1, 0

Offset: 0

Paul Barry, Jul 24 2005

Row sums are C(3;n), A064063. Inverse is A110517. Diagonal sums are A110525.

			Rows begin
  1;
  0,    1;
  0,    3,    1;
  0,   18,    6,    1;
  0,  135,   45,    9,    1;
  0, 1134,  378,   81,   12,    1;
  ...
Production matrix begins:
  0,   1;
  0,   3,   1;
  0,   9,   3,   1;
  0,  27,   9,   3,   1;
  0,  81,  27,   9,   3,   1;
  0, 243,  81,  27,   9,   3,   1;
  ... - _Philippe Deléham_, Sep 23 2014

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0] := 1; T[0, k_] := 0; T[n_, k_] := (k/n)*3^(n - k)*Binomial[2*n - k - 1, n - k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

concat([1], for(n=1,10, for(k=0,n, print1((k/n)*3^(n-k)*binomial(2*n-k-1,n-k), ", ")))) \\ G. C. Greubel, Aug 29 2017

Number triangle: T(0,k) = 0^k, T(n,k) = (k/n)*C(2n-k-1, n-k)*3^(n-k), n > 0, k > 0.
T(n,k) = A106566(n,k)*3^(n-k). - Philippe Deléham, Nov 08 2007
Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 3, 3, 3, 3, 3, 3, 3, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 23 2014

A132364 Expansion of 1/(1-x^2*c(x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 1, 3, 7, 20, 59, 184, 593, 1964, 6642, 22845, 79667, 281037, 1001092, 3595865, 13009673, 47366251, 173415176, 638044203, 2357941142, 8748646386, 32576869203, 121701491701, 456012458965, 1713339737086

Offset: 0

Philippe Deléham, Nov 08 2007

nonn

Diagonal sums of A106566.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.

Cf. A000108, A030238, A106566

a[0] := 1; a[n_] := Sum[(k/(n - k))*Binomial[2*n - 3*k - 1, n - 2*k], {k, 0, Floor[n/2]}]; Table[a[n], {n,0,25}] (* G. C. Greubel, Oct 19 2016 *)

c(x) = (1 - sqrt(1 - 4*x)) / (2*x); \\ A000108
my(x='x+O('x^30)); Vec(1/(1-x^2*c(x))) \\ Michel Marcus, Nov 13 2022

a(0)=1, a(n) = Sum_{k=0..floor(n/2)} (k/(n-k))*C(2*n-3*k-1,n-2*k), n>0.
G.f.: (2-x-x*sqrt(1-4*x))/(2-2*x+2*x^3). - Philippe Deléham, Feb 24 2013
Conjecture: +(-n+1)*a(n) +(5*n-11)*a(n-1) +2*(-2*n+5)*a(n-2) +(-n+1)*a(n-3) +2*(2*n-5)*a(n-4)=0. - R. J. Mathar, Aug 28 2015
a(n) ~ 2^(2*n + 2) / (49 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2022

Typo in a(n) term corrected Johannes W. Meijer, Sep 13 2010

A158826 Third iteration of x*C(x) where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 3, 12, 54, 260, 1310, 6824, 36478, 199094, 1105478, 6227712, 35520498, 204773400, 1191572004, 6990859416, 41313818217, 245735825082, 1470125583756, 8840948601024, 53417237877396, 324123222435804, 1974317194619712

Offset: 1

Paul D. Hanna, Mar 28 2009

nonn

Series reversion of x - 3*x^2 + 6*x^3 - 9*x^4 + 10*x^5 - 8*x^6 + 4*x^7 - x^8. - Benedict W. J. Irwin, Oct 19 2016

Column 1 of A106566^3 (see Barry, Section 3). - Peter Bala, Apr 11 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5, pp. 1-24.
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A121988 (2nd), A158825, A158827 (4th), A158828, A158829.

max = 22; c[x_] := Sum[ CatalanNumber[n]*x^n, {n, 0, max}]; f[x_] := x*c[x]; CoefficientList[ Series[ f@f@f@x, {x, 0, max}], x] // Rest (* Jean-François Alcover, Jan 24 2013 *)
Rest@CoefficientList[InverseSeries[x-3x^2+6x^3-9x^4+10x^5-8x^6+4x^7-x^8+O[x]^30], x] (* Benedict W. J. Irwin, Oct 19 2016 *)

a(n):=sum(binomial(2*k-2,k-1)*sum(binomial(-k+2*i-1,i-1)*binomial(2*n-i-1,n-1),i,k,n),k,1,n)/n; /* Vladimir Kruchinin, Jan 24 2013 */

a(n)=local(F=serreverse(x-x^2+O(x^(n+1))),G=x); for(i=1,3,G=subst(F,x,G)); polcoeff(G,n)

from sympy import binomial as C
def a(n):
    return sum(C(2*k - 2, k - 1) * sum(C(-k + 2*i - 1, i - 1) * C(2*n - i - 1, n - 1) for i in range(k, n + 1)) for k in range(1, n + 1)) / n
[a(n) for n in range(1, 51)] # Indranil Ghosh, Apr 12 2017

a(n) = (1/n)*Sum_{k=1..n} [ binomial(2*k-2,k-1)*Sum_{i=k..n}( binomial(-k+2*i-1,i-1)*binomial(2*n-i-1,n-1) ) ]. - Vladimir Kruchinin, Jan 24 2013
G.f.: (1 - sqrt(-1 + 2*sqrt(-1 + 2*sqrt(1 - 4*x))))/2. - Benedict W. J. Irwin, Oct 19 2016
a(n) ~ 2^(8*n - 3) / (sqrt(5*Pi) * n^(3/2) * 39^(n - 1/2)). - Vaclav Kotesovec, Jul 20 2019
Conjecture D-finite with recurrence 1053*n*(n-1)*(n-2)*(n-3)*a(n) -36*(n-1)*(n-2)*(n-3)*(634*n-1367)*a(n-1) +24*(n-2)*(n-3)*(7966*n^2-43500*n+61181)*a(n-2) -8*(n-3)*(96128*n^3-957424*n^2+3221878*n-3665189)*a(n-3) +16*(91904*n^4-1446528*n^3+8575792*n^2-22703688*n+22652013)*a(n-4) -256*(8*n-35)*(8*n-41)*(8*n-39)*(8*n-37)*a(n-5)=0. - R. J. Mathar, Aug 30 2021

A307496 Expansion of Product_{k>=1} (1 + ((1 - sqrt(1 - 4*x))/2)^k).

Original entry on oeis.org

1, 1, 2, 6, 18, 57, 187, 629, 2156, 7502, 26427, 94053, 337653, 1221260, 4445892, 16277089, 59893052, 221370725, 821499759, 3059620076, 11432831745, 42848889316, 161032785057, 606710026659, 2291156662259, 8670805904186, 32879697168622, 124910667052026, 475357627716839, 1811931609379926

Offset: 0

Ilya Gutkovskiy, Apr 10 2019

nonn

Catalan transform of A000009 (number of partitions into distinct parts).
From Peter Bala, Jan 27 2020: (Start)
This sequence is the main diagonal of the lower triangular array formed by putting A000009 in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.
    1
    1    1
    1    2    2
    2    4    6    6
    2    6   12   18   18
    3    9   21   39   57   57
  ...
Alternatively, the sequence can be obtained by multiplying A000009 by the array A106566.
(End)

Cf. A000009, A100100, A286955, A106566, A307495.

nmax = 29; CoefficientList[Series[Product[(1 + ((1 - Sqrt[1 - 4 x])/2)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Product[1/(1 - ((1 - Sqrt[1 - 4 x])/2)^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k PartitionsQ[k], {k, n}], {n, 29}]]

G.f.: Product_{k>=1}  1/(1 - ((1 - sqrt(1 - 4*x))/2)^(2*k-1)).
Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000009.
a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*A000009(k) for n > 0.
a(n) ~ c * 4^n / n^(3/2), where c = 1/sqrt(Pi) * Sum_{k>=0} k*A000009(k)/2^(k+1) = 1.12333545392999500455446757207126193339498222754079045166328600452997969... - Vaclav Kotesovec, Jan 28 2020, extended Aug 01 2022

A143464 Catalan transform of the Pell sequence.

Original entry on oeis.org

0, 1, 3, 11, 42, 164, 649, 2591, 10408, 41998, 170050, 690370, 2808714, 11446642, 46715469, 190876527, 780679200, 3195628806, 13090353594, 53655587034, 220045073988, 902842397664, 3705876933930, 15216954519222, 62503485455208

Offset: 0

Sergio Falcon, Oct 24 2008

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
Sergio Falcón, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.
Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals 2007; 33(1): 38-49.
Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
Merve Taştan and Engin Özkan, Catalan transform of the k-Pell, k-Pell-Lucas and modified k-Pell sequence, Notes on Num. Theory and Disc. Math. (2021) Vol. 27, No. 1, 198-207.

Cf. A000035, A000129, A016116, A039599, A106566, A109262.

a[n_]:= a[n]= If[n==0, 0, Sum[i*Binomial[2n-i,n-i]*Fibonacci[i,2]/(2n-i), {i,n}]];
Table[a[n], {n,0,30}] (* modified by G. C. Greubel, May 31 2022 *)

my(x='x+O('x^66)); concat([0],Vec((1-5*x-(1+x)*sqrt(1-4*x))/(2*x^2+16*x-4))) \\ Joerg Arndt, May 01 2013

def Pell(n): return round( ((1+sqrt(2))^n - (1-sqrt(2))^n)/(2*sqrt(2)) )
[0]+[(1/n)*sum(k*binomial(2*n-k-1, n-1)*Pell(k) for k in (1..n)) for n in (1..30)] # G. C. Greubel, May 31 2022

a(n) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n)*Pell(j), with a(0) = 0.
From Philippe Deléham, Oct 28 2008: (Start)
a(n) = Sum_{k=0..n} A106566(n,k)*A000129(k).
a(n) = Sum_{k=0..n} A039599(n,k)*A000035(k)*A016116(k). (End)
G.f.: ((1+x)*sqrt(1-4*x) - (1-5*x))/(2*(2 - 8*x - x^2)). - Mark van Hoeij, May 01 2013
a(n) = (1/(2*sqrt(2)))*Catalan(n-1)*Sum_{j=0..1} ((-1)^j + sqrt(2)) * Hypergeometric2F1([2,1-n], [2*(1-n)], 1+(-1)^j*sqrt(2)) - [n=0]/2. - G. C. Greubel, May 31 2022
a(n) ~ (1 + sqrt(2))^(2*n - 1) / 2^(2 + n/2). - Vaclav Kotesovec, May 31 2022

Offset corrected by Philippe Deléham, Oct 28 2008

A084868 Main diagonal of symmetric square table A084867, in which the antidiagonal sums (A006012) form the first row shifted left.

Original entry on oeis.org

1, 2, 8, 36, 168, 796, 3800, 18216, 87536, 421292, 2029592, 9784088, 47187536, 227651352, 1098523504, 5301727824, 25590307552, 123529362124, 596337248024, 2878947861432, 13899229883024, 67105641925064, 323993230750672

Offset: 0

Paul D. Hanna, Jun 10 2003, Jun 11 2003

The Hankel transform (see A001906 for definition) of this sequence is A000302 (powers of 4): 1, 4, 16, 64, 256, 1024, ... - Philippe Deléham, Aug 17 2005

			1 + 2*x + 8*x^2 + 36*x^3 + 168*x^4 + 796*x^5 + 3800*x^6 + 18216*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vincent Pilaud, V Pons, Permutrees, arXiv preprint arXiv:1606.09643, 2016

Cf. A006012, A011782, A028329, A084867, A026671, A081696.

1/(1-x/(sqrt(1/4-x))): series(%,x,23): seq(coeff(%,x,n),n=0..22); # Peter Luschny, Feb 06 2017

Table[SeriesCoefficient[((1-4*x)+2*x*Sqrt[1-4*x])/(1-4*x-4*x^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

{a(n) = if( n<0, 0, polcoeff((1 - 4*x + 2*x * sqrt(1 - 4*x + x * O(x^n))) /(1 - 4*x - 4*x^2), n))} /* Michael Somos, Jan 05 2012 */

Differential equation: (16*x^3 + 12*x^2 - 8*x + 1) * x*(d/dx)A(x) + (8x^3 - 12*x^2 + 6*x - 1) * A(x) + (8x^2 - 6*x + 1) = 0.
G.f.: ((1 - 4*x) + 2*x * sqrt(1 - 4*x)) / (1 - 4*x - 4*x^2). a(n) * (n-1) = a(n-1) * (8*n - 14) - a(n-2) * 12*(n-3) - a(n-3) * 8*(2*n - 5), n > 2. Hankel number wall zig-zag diagonal is A011782. - Michael Somos, Sep 14 2003
INVERT transform of A028329 (offset 1). - Michael Somos, Jan 05 2012
G.f.: (1-2*x*f(x))/(1-2*x*f(x)-2*x) where f(x) is the g.f. of A000108 (Catalan numbers). - Philippe Deléham, Jan 30 2012
a(n) ~ (1-1/sqrt(2))*(2+2*sqrt(2))^n. - Vaclav Kotesovec, Oct 14 2012
From Peter Bala, Feb 05 2017: (Start)
G.f: sqrt(1 - 4*x)/(sqrt(1 - 4*x) - 2*x) =  1/(1 - 2*x/(1 - 2*x/(1 - x/(1 - x/(1 - x/(1 - ...)))))) (continued fraction).  Cf. A026671, A081696.
Catalan transform of A006012, that is, equals A106566*A006012, as noted by R. J. Mathar. (End)

A102625 Triangle read by rows: T(n,k) is the sum of the weights of all vertices labeled k at depth n in the Catalan tree (1 <= k <= n+1, n >= 0).

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 15, 30, 36, 24, 105, 210, 270, 240, 120, 945, 1890, 2520, 2520, 1800, 720, 10395, 20790, 28350, 30240, 25200, 15120, 5040, 135135, 270270, 374220, 415800, 378000, 272160, 141120, 40320, 2027025, 4054050, 5675670, 6486480

Offset: 0

Emeric Deutsch, Jan 31 2005

The Catalan tree is defined as follows: the root is labeled 1 and each vertex labeled i has i+1 children labeled 1,2,...,i+1. The weight of a vertex v is the product of all labels on the path from the root to v. Row n contains n+1 terms. Row sums and column 1 yield the double factorials (A001147). T(n,n+1)=(n+1)!, T(n,n)=n(n+1)!/2 (A001286; Lah numbers).
This table counts permutations of the multiset {1,1,2,2,...,n,n} satisfying the condition "the first appearance of i + 1 follows the first appearance of i" by the position of the first appearance of n. Specifically, T(n+1,k) is the number of such permutations for which n first occurs in position 2n+1-k. For example, with n=2 and k=1, T(3,1)=6 counts 121323, 121332, 122313, 122331, 112323, 112332. - David Callan, Nov 29 2007
T(n+1,k) is also the number of rooted complete binary forests with n labeled leaves and k labeled roots. This follows by comparing exponential generating functions; see Example 5.2.6 and Proposition 5.1.3 of Stanley's "Enumerative Combinatorics 2." - Timothy Y. Chow, Mar 28 2017

			Triangle starts:
   1;
   1,  2;
   3,  6,  6;
  15, 30, 36, 24;
  ...
Production matrix begins:
1, 2
1, 2, 3
1, 2, 3, 4
1, 2, 3, 4, 5
1, 2, 3, 4, 5, 6
1, 2, 3, 4, 5, 6, 7
1, 2, 3, 4, 5, 6, 7, 8
1, 2, 3, 4, 5, 6, 7, 8, 9
... - _Philippe Deléham_, Sep 30 2014
From _Peter Bala_, Apr 16 2017: (Start)
The Catalan tree starts          o1
                                / \
                               /   \
                              /     \
                             /       \
                            /         \
                           o1          o2
                          / \         /|\
                         /   \       / | \
                        /     \     /  |  \
                       o1      o2  o1  o2  o3
Level 2:
2 vertices labeled 1: total weight 1x1x1 + 1x2x1 = 3
2 vertices labeled 2: total weight 2x1x1 + 2x2x1 = 6
1 vertex labeled 3:   total weight 3x2x1         = 6
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
M. R. T. Dale and J. W. Moon, The permuted analogues of three Catalan sets, J. Stat. Plan. Inf. 34 (1) (1993) 75-87 Table 1
S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
Jiaxi Lu and Yuanzhe Ding, A skeleton model to enumerate standard puzzle sequences, arXiv:2106.09471 [math.CO], 2021.
Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

Cf. A001147 (row sums), A001286, A001497, A039683, A132382, A145271, A000108, A033184.

A102625:=proc(n,k) if k<=n+1 then k*(2*n-k+1)!/2^(n-k+1)/(n-k+1)! else 0 fi end proc:
for n from 0 to 8 do seq(A102625(n,k),k=1..n+1) od; # yields sequence in triangular form

t[n_, k_] := k*(2n-k+1)!/(2^(n-k+1)*(n-k+1)!); Table[t[n, k], {n, 0, 8}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Jan 21 2013 *)

{T(n, k) = my(m = n-k+1); if( k<1 || k>n+1, 0, k * (n+m)! / (2^m * m!))}; /* Michael Somos, Aug 16 2016 */

T(n,k) = k*(2*n-k+1)!/[2^(n-k+1)*(n-k+1)!] (1 <= k <= n+1).
From Tom Copeland, Nov 11 2007: (Start)
Bivariate G.F.: exp[P(.,t)*x] = D_x {1 - [g(x)/(1+t*g(x))]} = 1 / {(1+g(x))*[1+t*g(x)]^2}, where g(x) = sqrt(1-2*x) - 1 and P(n,t) = Sum_{k=0..n} T(n,k) * t^k.
Also D_x g(x) = -(1-2*x)^(-1/2) = -exp[x*A001147(.)] = -exp[x *(2*(.)-1)!! ], so the coefficients of x^n/n! in the expansion of g(x) are -(2*(n-1)-1)!! = -A001147(n-1) for n > 0.
See A132382 for an array which is essentially the revert from which this G.f. may be derived and for connections to other arrays. (End)
E.g.f.: 1/(1 - x + x*sqrt(1-2*z)) = 1 + x*z + (x+2*x^2)*z^2/2! + (3*x+6*x^2+6*x^3)*z^3/3! + .... T(n,k) gives the number of plane recursive trees on n+2 nodes where the root has degree k (Bergeron et al., Corollary 5). - Peter Bala, Jul 09 2012
From Peter Bala, Jul 09 2014: (Start)
T(n,k) = k!*A001497(n,k) modulo offset differences.
The n-th row polynomial R(n,x) = (-1)^n/(x - 1)*( Sum_{k = 1..infinity} k*(k - 2)*...*(k - 2*n)*(x/(x - 1))^k ). Cf. the Dobinski-type formula for the row polynomials of A001497. (End)
From Tom Copeland, Aug 06 2016: (Start)
From the 2007 formulas above, an alternate g.f. for this entry is GF(x,t) = -g(x) / [1 + t*g(x)] = x + (1 + 2*t)*x^2/2! + (3 + 6*t + 6*t^2)*x^3/3! + ... with compositional inverse GFinv(x,t) = {1 - [1 - x / (1+t*x)]^2} / 2 = -(1/2)[x / (1+t*x)]^2 + x / (1+t*x) = Sum_{n>0} (-1)^(n+1) [(n-1)/2*t^(n-2) + t^(n-1)]*x^n, a series containing the Lah numbers A001286 when expressed as an e.g.f.
From A145271, with K(x,t) = 1 / dGinv(x,t)/dx = 1 + (1+2*t) x + (1+t+t^2) x^2 + x^3 / [1-(1-t)*x], then [K(x,t) d/dx]^n x evaluated at x=0 gives the n-th row polynomial of this entry.
Since the reciprocal of Bala's e.g.f. above generates a shifted, signed A001147, for the polynomials P(n,t) generated by Bala's e.g.f., umbrally (P(.,t) + a.)^n = 0 for n > 0 with a_0 = 1 and a_n = -t * A001147(n-1) for n > 0. E.g., (P(.,t) + a.)^2 = a_0 * P(2,t) + 2 a_1 * P(1,t) + a_2 * P(0,t) = 1 * (t + 2*t^2) + 2 * -t * t + -t * 1 = 0. (End)
From Peter Bala, Apr 16 2017: (Start)
T(n,k) = k*T(n-1,k-1) + (2*n - k)*T(n-1,k).
E.g.f.: t*x*c(x/2)/(1 - t*x*c(x/2)) = t*x + (t + 2*t^2)*x^2/2! + (3*t + 6*t^2 + 6*t^2)*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. for the Catalan numbers A000108. Note that the related g.f. t*x*c(x)/(1 - t*x*c(x)) is the o.g.f. for A033184 (essentially the same as the Riordan array A106566) and enumerates the number of vertices labeled k on the n_th level of the Catalan tree (k >= 1, n >= 0). (End)

A127543 Triangle T(n,k), 0<=k<=n, read by rows given by :[ -1,1,1,1,1,1,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, -1, 1, -1, 1, -2, 0, 2, -1, 1, -6, 2, 1, 3, -1, 1, -18, 5, 7, 2, 4, -1, 1, -57, 17, 19, 13, 3, 5, -1, 1, -186, 56, 64, 36, 20, 4, 6, -1, 1, -622, 190, 212, 124, 56, 28, 5, 7, -1, 1, -2120, 654, 722, 416, 198, 79, 37, 6, 8, -1, 1, -7338, 2282, 2494, 1434, 673, 287, 105, 47, 7, 9, -1, 1

Offset: 0

Philippe Deléham, Apr 01 2007

Riordan array (2/(3-sqrt(1-4*x)), (1-sqrt(1-4*x))/(3-sqrt(1-4*x))). - Philippe Deléham, Jan 27 2014

			Triangle begins:
    1;
   -1,  1;
    0, -1,  1;
   -1,  1, -1,  1;
   -2,  0,  2, -1,  1;
   -6,  2,  1,  3, -1,  1;
  -18,  5,  7,  2,  4, -1,  1;
  -57, 17, 19, 13,  3,  5, -1, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

A065600[n_, k_]:= If[k==n, 1, Sum[j*Binomial[k+j, j]*Binomial[2*(n-k-j), n-k]/(n-k-j), {j,0, Floor[(n-k)/2]}]];
A127543[n_, k_]:= A065600[n-1,k-1] - A065600[n-1,k];
Table[A127543[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 17 2021 *)

def A065600(n,k): return 1 if (k==n) else sum( j*binomial(k+j, j)*binomial(2*(n-k-j), n-k)/(n-k-j) for j in (0..(n-k)//2) )
def A127543(n,k): return A065600(n-1, k-1) - A065600(n-1, k)
flatten([[A127543(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 17 2021

T(n,k) = A065600(n-1,k-1) - A065600(n-1,k).
Sum_{k=0..n} T(n,k)*x^k = A127053(n), A126985(n), A127016(n), A127017(n), A126987(n), A126986(n), A126982(n), A126984(n), A126983(n), A000007(n), A000108(n), A000984(n), A007854(n), A076035(n), A076036(n), A127628(n), A126694(n), A115970(n) for n= -8,-7,...,8,9 respectively.
Sum_{j>=0} T(n,j)*A007318(j,k) = A106566(n,k).
Sum_{j>=0} T(n,j)*A038207(j,k) = A039599(n,k).
Sum_{j>=0} T(n,j)*A027465(j,k) = A116395(n,k).

cp's OEIS Frontend

A127628 G.f. 1/(1-6*x*c(x)) where c(x) is the g.f. of A000108.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A110510 Riordan array (1, x*c(2x)), c(x) the g.f. of A000108.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

PARI

Formula

A110518 Riordan array (1, x*c(3x)), c(x) the g.f. of A000108.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

PARI

Formula

A132364 Expansion of 1/(1-x^2*c(x)), c(x) the g.f. of A000108.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A158826 Third iteration of x*C(x) where C(x) is the Catalan function (A000108).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Python

Formula

A307496 Expansion of Product_{k>=1} (1 + ((1 - sqrt(1 - 4*x))/2)^k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A143464 Catalan transform of the Pell sequence.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

SageMath

Formula

A127628 G.f. 1/(1-6xc(x)) where c(x) is the g.f. of A000108.