A080247 -id:A080247 - cp's OEIS frontend

A330801 a(n) = A080247(2*n, n), the central values of the Big-Schröder triangle.

1, 4, 30, 264, 2490, 24396, 244790, 2496528, 25763058, 268243860, 2812481870, 29653804824, 314097641130, 3339741725404, 35626286189670, 381098437754912, 4086504567333858, 43912100376527652, 472743964145437310, 5097853987059017000, 55054474579787825562

Offset: 0

Peter Luschny, Jan 02 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 0..950

Row sums of A330798.

Cf. A080247.

A330801:= func< n | ((n+1)/(2*n+1))*(&+[Binomial(2*n+1, n+j+1)*Binomial(2*n+j, j): j in [0..n]]) >;
[A330801(n): n in [0..40]]; // G. C. Greubel, May 03 2023

a := n -> ((n+1)/(2*n+1))*binomial(2*n+1, n+1)*hypergeom([-n, 2*n+1], [n+2], -1):
seq(simplify(a(n)), n=0..20);
# Alternative:
alias(C = binomial):
a := n -> ((n+1)/(2*n+1))*add(C(2*n+1, n+j+1)*C(2*n+j, j), j=0..n):
seq(a(n), n=0..20);

a[n_]:= (1/Sqrt[Pi]) 4^n (1 + n) Gamma[1/2 + n] Hypergeometric2F1Regularized[-n, 1 + 2 n, 2 + n, -1]; Table[a[n], {n, 0, 20}]

def A330801(n) -> int:
    s = sum( binomial(2 * n + 1, n + j + 1) * binomial(2 * n + j, j)
        for j in range(n + 1) )
    return (s * (n + 1)) // (2 * n + 1)
print([A330801(n) for n in range(41)])  # G. C. Greubel, May 03 2023

a(n) = ((n+1)/(2*n+1))*Sum_{j=0..n} binomial(2*n+1, n+j+1) * binomial(2*n+j, j).
a(n) = ((n+1)/(2*n+1))*binomial(2*n+1, n+1)*hypergeom([-n, 2*n + 1], [n + 2], -1).
D-finite with recurrence 2*n*(2*n+1)*(7*n-13)*a(n) - (382*n^3 -983*n^2 +533*n -40)*a(n-1) + (n-2)*(786*n^2 -3290*n +3315)*a(n-2) + (2*n-5)*(37*n-39)*(n-3)*a(n-3) = 0. - R. J. Mathar, Jul 27 2022
a(n) ~ phi^(5*n + 1) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 03 2023

A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 16, 22, 1, 8, 30, 68, 90, 1, 10, 48, 146, 304, 394, 1, 12, 70, 264, 714, 1412, 1806, 1, 14, 96, 430, 1408, 3534, 6752, 8558, 1, 16, 126, 652, 2490, 7432, 17718, 33028, 41586, 1, 18, 160, 938, 4080, 14002, 39152, 89898, 164512, 206098

Offset: 1

N. J. A. Sloane

A106579 is in some ways a better version of this sequence, but since this was entered first it will be the main entry for this triangle.
The diagonals of this triangle are self-convolutions of the main diagonal A006318: 1, 2, 6, 22, 90, 394, 1806, ... . - Philippe Deléham, May 15 2005
From Johannes W. Meijer, Sep 22 2010, Jul 15 2013: (Start)
Note that for the terms T(n,k) of this triangle n indicates the column and k the row.
The triangle sums, see A180662, link Schroeder's triangle with several sequences, see the crossrefs. The mirror of this triangle is A080247.
Quite surprisingly the Kn1p sums, p >= 1, are all related to A026003 and crystal ball sequences for n-dimensional cubic lattices (triangle offset is 0): Kn11(n) = A026003(n), Kn12(n) = A026003(n+2) - 1, Kn13(n) = A026003(n+4) - A005408(n+3), Kn14(n) =  A026003(n+6) - A001844(n+4), Kn15(n) = A026003(n+8) - A001845(n+5), Kn16(n) = A026003(n+10) - A001846(n+6), Kn17(n) = A026003(n+12) - A001847(n+7), Kn18(n) = A026003(n+14) - A001848(n+8), Kn19(n) = A026003(n+16) - A001849(n+9), Kn110(n) = A026003(n+18) - A008417(n+10), Kn111(n) = A026003(n+20) - A008419(n+11), Kn112(n) = A026003(n+22) - A008421(n+12). (End)
T(n,k) is the number of normal semistandard Young tableaux with two columns, one of height k and one of height n. The recursion can be seen by performing jeu de taquin deletion on all instances of the smallest value. (If there are two instances of the smallest value, jeu de taquin deletion will always shorten the right column first and the left column second.) - Jacob Post, Jun 19 2018

			Triangle starts:
  1;
  1,    2;
  1,    4,    6;
  1,    6,   16,   22;
  1,    8,   30,   68,   90;
  1,   10,   48,  146,  304,  394;
  1,   12,   70,  264,  714, 1412, 1806;
  ... - _Joerg Arndt_, Sep 29 2013

T. D. Noe, Rows k = 1..50 of triangle, flattened
Henry Bottomley, Illustration of initial terms
Kevin Brown, Hipparchus on Compound Statements, 1994-2010. - _Johannes W. Meijer_, Sep 22 2010
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
J. W. Meijer, Famous numbers on a chessboard, Acta Nova, Volume 4, No.4, December 2010. pp. 589-598.
J. M. Oh, An explicit formula for the number of fuzzy subgroups of a finite abelian p-group of rank two, Iranian Journal of Fuzzy Systems, Dec 2013, Vol. 10 Issue 6, pp. 125-135.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
S. Samieinia, The number of continuous curves in digital geometry, Port. Math. 67 (1) (2010) 75-89, last table.
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), Article 03.1.5, 19 pp.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Essentially same triangle as A080247 and A080245 but with rows read in reversed order. Also essentially the same triangle as A106579.
Cf. A000007, A006318, A006319, A006320, A006321, A008288, A080243.
Cf. A084938, A103885, A238112, A330801.
Cf. A001003 (row sums), A026003 (antidiagonal sums).
Triangle sums (see the comments): A001003 (Row1, Row2), A026003 (Kn1p, p >= 1), A006603 (Kn21), A227504 (Kn22), A227505 (Kn23), A006603(2*n) (Kn3), A001850 (Kn4), A227506 (Fi1), A010683 (Fi2).

a033877 n k = a033877_tabl !! n !! k
a033877_row n = a033877_tabl !! n
a033877_tabl = iterate
   (\row -> scanl1 (+) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Apr 17 2013

function t(n,k)
  if k le 0 or k gt n then return 0;
  elif k eq 1 then return 1;
  else return t(n,k-1) + t(n-1,k-1) + t(n-1,k);
  end if;
end function;
[t(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 23 2023

T := proc(n, k) option remember; if n=1 then return(1) fi; if kJohannes W. Meijer, Sep 22 2010, revised Jul 17 2013

Mathematica
```
T[1, ]:= 1; T[n, k_]/;(k
				
```

def A033877_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^k*prec(n, n-k) for k in (0..n-1)]
for n in (1..10): print(A033877_row(n)) # Peter Luschny, Mar 16 2016

@CachedFunction
def t(n, k): # t = A033847
    if (k<0 or k>n): return 0
    elif (k==1): return 1
    else: return t(n, k-1) + t(n-1, k-1) + t(n-1, k)
flatten([[t(n,k) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 23 2023

As an upper right triangle: a(n, k) = a(n, k-1) + a(n-1, k-1) + a(n-1, k) if k >= n >= 0 and a(n, k) = 0 otherwise.
G.f.: Sum T(n, k)*x^n*y^k = (1-x*y-(x^2*y^2-6*x*y+1)^(1/2)) / (x*(2*y+x*y-1+(x^2*y^2-6*x*y+1)^(1/2))). - Vladeta Jovovic, Feb 16 2003
Another version of A000007 DELTA [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] = 1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, ..., where DELTA is Deléham's operator defined in A084938.
Sum_{n=1..floor((k+1)/2)} T(n+p-1, k-n+p) = A026003(2*p+k-3) - A008288(2*p+k-3, p-2), p >= 2, k >= 1. - Johannes W. Meijer, Sep 28 2013
From G. C. Greubel, Mar 23 2023: (Start)
(t(n, k) as a lower triangle)
t(n, k) = t(n, k-1) + t(n-1, k-1) + t(n-1, k) with t(n, 1) = 1.
t(n, n) = A006318(n-1).
t(2*n-1, n) = A330801(n-1).
t(2*n-2, n) = A103885(n-1), n > 1.
Sum_{k=1..n-1} t(n, k) = A238112(n), n > 1.
Sum_{k=1..n} t(n, k) = A001003(n).
Sum_{k=1..n-1} (-1)^(k-1)*t(n, k) = (-1)^n*A001003(n-1), n > 1.
Sum_{k=1..n} (-1)^(k-1)*t(n, k) = A080243(n-1).
Sum_{k=1..floor((n+1)/2)} t(n-k+1, k) = A026003(n-1). (End)

More terms from David W. Wilson

A080246 Signed version of A035607.

Original entry on oeis.org

1, -2, 1, 2, -4, 1, -2, 8, -6, 1, 2, -12, 18, -8, 1, -2, 16, -38, 32, -10, 1, 2, -20, 66, -88, 50, -12, 1, -2, 24, -102, 192, -170, 72, -14, 1, 2, -28, 146, -360, 450, -292, 98, -16, 1, -2, 32, -198, 608, -1002, 912, -462, 128, -18, 1, 2, -36, 258, -952, 1970, -2364

Offset: 0

Paul Barry, Feb 15 2003

Written as lower triangular matrix this has inverse A080247. Row sums are (1,-1,-1,1,1,-1,-1,1,1,...) Diagonal sums are signed tribonacci numbers A078042

Riordan array((1-x)/(1+x), x*(1-x)/(1+x)). - Philippe Deléham, Jan 05 2014

			Rows are {1}, {-2,1}, {2,-4,1}, {-2,8,-6,1}, ...

Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292, 2013
Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv preprint arXiv:1105.3713, 2011.

Cf. A035607, A080247.

Columns are generated by (1-x)^k/(1+x)^k
T(n,k)=(-1)^(n+k)*A113413(n,k). - Philippe Deléham, Jan 05 2014
T(n,k)=T(n-1,k-1)-T(n-1,k)-T(n-2,k-1), T(0,0)=1, T(1,0)=-2, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 05 2014

A330802 Evaluation of the Big-Schröder polynomials at 1/2 and normalized with 2^n.

Original entry on oeis.org

1, 5, 33, 253, 2121, 18853, 174609, 1667021, 16290969, 162171445, 1638732129, 16765758429, 173325794409, 1807840791237, 19001320087473, 201050792435949, 2139811906460985, 22892988893079637, 246061004607915777, 2655768423781296893, 28771902274699214601

Offset: 0

Peter Luschny, Jan 02 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..941

Cf. A080247, A330803, A330799, A330800.

a := proc(n) option remember; if n < 3 then return [1, 5, 33][n+1] fi;
((24 - 12*n)*a(n-3) + (32*n - 10)*a(n-2) + (9*n - 9)*a(n-1))/(n+1) end:
seq(a(n), n=0..20);
# Alternative:
gf := 2/(1 - 4*x + sqrt(1 + 4*(x - 3)*x)):
ser := series(gf, x, 24):
seq(coeff(ser, x, n), n=0..20);
# Or:
series((x - x^2)/(3*x^2 + 4*x + 1), x, 24):
gfun:-seriestoseries(%, 'revogf'):
convert(%, polynom) / x: seq(coeff(%, x, n), n=0..20);

A080247[n_, k_] := (k+1)*Sum[2^m*Binomial[n+1, m]*Binomial[n-k-1, n-k-m], {m, 0, n-k}]/(n+1);
a[n_] := 2^n*Sum[A080247[n, k]/2^k , {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 22 2023 *)

N=20; x='x+O('x^N); Vec(2/(1-4*x+sqrt(1+4*(x-3)*x))) \\ Seiichi Manyama, Feb 03 2020

R. = PowerSeriesRing(QQ)
f = (x - x^2)/(3*x^2 + 4*x + 1)
f.reverse().shift(-1).list()

a(n) = 2^n*Sum_{k=0..n} A080247(n,k)/2^k.
a(n) = ((24 - 12*n)*a(n-3) + (32*n - 10)*a(n-2) + (9*n - 9)*a(n-1))/(n + 1).
a(n) = [x^n] 2/(1 - 4*x + sqrt(1 + 4*(x - 3)*x)).
a(n) = [x^n] reverse((x - x^2)/(3*x^2 + 4*x + 1))/x.
a(n) ~ 2^(n + 5/4) * (1 + sqrt(2))^(2*n-1) / (sqrt(Pi) * (57 - 40*sqrt(2)) * n^(3/2)). - Vaclav Kotesovec, Oct 22 2023

A330803 Evaluation of the Big-Schröder polynomials at -1/2 and normalized with (-2)^n.

Original entry on oeis.org

1, -3, 17, -123, 1001, -8739, 79969, -756939, 7349657, -72798003, 732681489, -7471545435, 77031538377, -801616570947, 8408819677377, -88821190791915, 943928491520249, -10085451034660947, 108275140773938545, -1167408859459660923, 12635538801834255401

Offset: 0

Peter Luschny, Jan 02 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..941

Cf. A080247, A330802, A330799, A330800.

a := proc(n) option remember; if n < 3 then return [1, -3, 17][n+1] fi;
((8 - 4*n)*a(n-3) + (30 - 24*n)*a(n-2) + (17 - 37*n)*a(n-1))/(3*n + 3) end:
seq(a(n), n=0..20);
# Alternative:
gf := 2/(1 + sqrt(1 + 4*x*(x + 3))):
ser := series(gf, x, 24):
seq(coeff(ser, x, n), n=0..20);
# Or:
series((3*x^2 + x)/(1 - x^2), x, 24):
gfun:-seriestoseries(%, 'revogf'):
convert(%, polynom) / x: seq(coeff(%, x, n), n=0..20);

N=20; x='x+O('x^N); Vec(2/(1+sqrt(1+4*x*(x+3)))) \\ Seiichi Manyama, Feb 03 2020

R. = PowerSeriesRing(QQ)
f = (3*x^2 + x)/(1 - x^2)
f.reverse().shift(-1).list()

a(n) = (-2)^n*Sum_{k=0..n} A080247(n,k)/(-2)^k.
a(n) = ((8 - 4*n)*a(n-3) + (30 - 24*n)*a(n-2) + (17 - 37*n)*a(n-1))/(3*n + 3).
a(n) = [x^n] 2/(1 + sqrt(1 + 4*x*(x + 3))).
a(n) = [x^n] reverse((3*x^2 + x)/(1 - x^2))/x.

A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 22, 16, 6, 1, 0, 90, 68, 30, 8, 1, 0, 394, 304, 146, 48, 10, 1, 0, 1806, 1412, 714, 264, 70, 12, 1, 0, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 0, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1, 0, 206098, 164512, 89898, 39152, 14002, 4080, 938, 160, 18, 1

Offset: 0

Philippe Deléham, Sep 18 2006

Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 2, 1, 2, 1, ...] DELTA [1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . Inverse is Riordan array (1, x*(1-x)/(1+x)).

T(n, r) gives the number of [0,r]-covering hierarchies with n segments terminating at r (see Kreweras work). - Michel Marcus, Nov 22 2014

			Triangle begins:
  1;
  0,    1:
  0,    2,    1;
  0,    6,    4,    1;
  0,   22,   16,    6,    1;
  0,   90,   68,   30,    8,   1;
  0,  394,  304,  146,   48,  10,  1;
  0, 1806, 1412,  714,  264,  70, 12,  1;
  0, 8558, 6752, 3534, 1408, 430, 96, 14, 1;
Production matrix is:
  0...1
  0...2...1
  0...2...2...1
  0...2...2...2...1
  0...2...2...2...2...1
  0...2...2...2...2...2...1
  0...2...2...2...2...2...2...1
  0...2...2...2...2...2...2...2...1
  0...2...2...2...2...2...2...2...2...1
  ... - _Philippe Deléham_, Feb 09 2014

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), see page 15.

Another version : A080247, A080245, A033877.
Columns: A000007, A006318, A006319, A006320, A006321.
Diagonals: A000012, A005843, A054000.
Sums include: A001003 (row and alternating sign), A006603 (diagonal).
Cf. A103885.

function T(n,k) // T = A122538
  if k eq 0 then return 0^n;
  elif k eq n then return 1;
  else return T(n-1,k-1) + T(n-1,k) + T(n,k+1);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 27 2024

T[n_, n_]= 1; T[, 0]= 0; T[n, k_]:= T[n, k]= T[n-1, k-1] + T[n-1, k] + T[n, k+1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jun 13 2019 *)

def A122538_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k)*prec(n, k) for k in (0..n)]
for n in (0..12): print(A122538_row(n)) # Peter Luschny, Mar 16 2016

T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1) if k > 0, with T(n, 0) = 0^n, and T(n, n) = 1.
Sum_{k=0..n} T(n, k) = A001003(n).
From G. C. Greubel, Oct 27 2024: (Start)
T(2*n, n) = A103885(n).
Sum_{k=0..n} (-1)^k*T(n, k) = -A001003(n-1).
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] + 0*[n=1] + A006603(n-2)*[n>1]. (End)

A183875 Triangle T(n,k) for A(x)^k=sum(n>=k T(n,k)x^n), where o.g.f. A(x) satisfies A(x)=(a+bx*A(x))/(c-dxA(x)), a=1,b=2,c=1,d=2.

Original entry on oeis.org

1, 4, 1, 24, 8, 1, 176, 64, 12, 1, 1440, 544, 120, 16, 1, 12608, 4864, 1168, 192, 20, 1, 115584, 45184, 11424, 2112, 280, 24, 1, 1095424, 432128, 113088, 22528, 3440, 384, 28, 1, 10646016, 4227584, 1133952, 237824, 39840, 5216, 504, 32, 1, 105522176, 42115072, 11506944, 2505728, 448064, 65280, 7504, 640, 36, 1

Offset: 1

Vladimir Kruchinin, Feb 12 2011

For o.g.f G(x), G(A(x,a,b,c,d))=g(0)+sum(n>0, sum(k=1..n, T(n,k,a,b,c,d)*g(k))x^n).
T(n,k,1,1,1,1)=A080247(n,k),
T(n,k,2,-1,1,1)=A108891(n,k),
T(n,k,1,-2,1,1)=A125692(n,k),
T(n,k,1,-3,1,1)=A125694(n,k),
T(n,k,-2,1,1,1)=A085403(n,k).

			1,
4,1,
24,8,1,
176,64,12,1,
1440,544,120,16,1,
12608,4864,1168,192,20,1,
115584,45184,11424,2112,280,24,1,
1095424,432128,113088,22528,3440,384,28,1,
10646016,4227584,1133952,237824,39840,5216,504,32,1,
105522176,42115072,11506944,2505728,448064,65280,7504,640,36,1

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

T[n_, k_, a_, b_, c_, d_] := k/n Sum[Binomial[n, n - k - i] a^(k + i) b^(n - k - i) Binomial[i + n - 1, n - 1] c^(-i - n) d^i, {i, 0, n - k}];
T[n_, k_] := T[n, k, 1, 2, 1, 2];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 08 2018, from formula *)

T(n,k,a,b,c,d):=k/n*sum(i=0..n-k, binomial(n,n-k-i)*a^(k+i)*b^(n-k-i)*binomial(i+n-1,n-1)*c^(-i-n)*d^i), a,b,c,d !=0, n>0.
T(n,k,1,2,1,2):=k/n*2^(n-k)*sum(i=0..n-k, binomial(n,n-k-i)*binomial(i+n-1,n-1)), n>0.
Conjecture: T(n,1) = A156017(n-1). - R. J. Mathar, Nov 14 2011

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A330801 a(n) = A080247(2*n, n), the central values of the Big-Schröder triangle.

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A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

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A080246 Signed version of A035607.

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A330802 Evaluation of the Big-Schröder polynomials at 1/2 and normalized with 2^n.

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A330803 Evaluation of the Big-Schröder polynomials at -1/2 and normalized with (-2)^n.

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A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.

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A183875 Triangle T(n,k) for A(x)^k=sum(n>=k T(n,k)*x^n), where o.g.f. A(x) satisfies A(x)=(a+b*x*A(x))/(c-d*x*A(x)), a=1,b=2,c=1,d=2.

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A183875 Triangle T(n,k) for A(x)^k=sum(n>=k T(n,k)x^n), where o.g.f. A(x) satisfies A(x)=(a+bx*A(x))/(c-dxA(x)), a=1,b=2,c=1,d=2.