A085478 -id:A085478 - cp's OEIS frontend

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

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).

A101919 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at even heights.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 8, 1, 1, 33, 42, 13, 1, 1, 88, 183, 102, 19, 1, 1, 232, 717, 624, 205, 26, 1, 1, 609, 2622, 3275, 1650, 366, 34, 1, 1, 1596, 9134, 15473, 11020, 3716, 602, 43, 1, 1, 4180, 30691, 67684, 64553, 30520, 7483, 932, 53, 1, 1, 10945, 100284, 279106

Offset: 0

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 humps. A hump is an up step U followed by 0 or more level steps H followed by a down step D. The T(3,2)=8 Schroeder paths of length 6 and having 2 humps are: H(UD)(UD), (UD)H(UD), (UD)(UD)H, (UD)(UHD), (UD)(UUDD), (UHD)(UD), (UUDD)(UD) and U(UD)(UD)D, the humps being shown between parentheses. Row sums are the large Schroeder numbers (A006318). Column 1 yields the odd-indexed Fibonacci numbers minus 1 (A027941). T(n,n-1)=A034856(n)=binomial(n + 1, 2) + n - 1.

Product A085478*A090181 (Morgan-Voyce times Narayana). [From Paul Barry, Jan 29 2009]

			T(3,2)=8 because we have HU'DU'D, U'DHU'D, U'DU'DH, U'DU'HD, U'DU'UDD, U'HDU'D, U'UDDU'D and U'UU'DDD, the up steps starting at an even height being shown with a prime sign.
Triangle begins:
1;
1,1;
1,4,1;
1,12,8,1;
1,33,42,13,1;

Cf. A006318, A027941, A034856, A101920.

G:=1/2/(-z+z^2)*(-1+z+t*z-z^2+sqrt(1-6*z-2*t*z+11*z^2+2*t*z^2-6*z^3+t^2*z^2-2*t*z^3+z^4)): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields the sequence in triangular form

G.f.=G=G(t, z) satisfies z(1-z)G^2-(1-z-tz+z^2)G+1-z=0.

G.f.: 1/(1-x-xy/(1-x-x/(1-x-xy/(1-x-xy/(1-x-x/(1-x-xy/(1-.... (continued fraction). [From Paul Barry, Jan 29 2009]

A144250 Eigentriangle, row sums = A125275, shifted.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 23, 1, 15, 70, 168, 207, 106, 1, 21, 140, 504, 1035, 1166, 567, 1, 28, 252, 1260, 3795, 6996, 7371, 3434

Offset: 0

Gary W. Adamson, Sep 16 2008

Row sums = A125273 shifted. A125273 = the eigensequence of triangle A085478.

Right border = A125273: (1, 1, 2, 6, 23, 106, 567, 3434,...). Sum of n-th row terms = rightmost term in next row.

			First few rows of the triangle =
1;
1, 1;
1, 3, 2;
1, 6, 10, 6;
1, 10, 30, 42, 23;
1, 15, 70, 168, 207, 106;
1, 21, 140, 504, 1035, 1166, 567;
...
Row 4 = (1, 10, 30, 42, 23) = termwise products of (1, 10, 15, 7, 1) and (1, 1, 2, 6, 23) = (1*1, 10*1, 15*2, 7*6, 1*23); where (1, 10, 15, 7, 1) = row 4 of triangle A085478. Q

Cf. A125273, A085478.

Triangle read by rows, T(n,k) = A085478(n,k) * A125273(k).

As infinite lower triangular matrices, A144250 = A085478 * (A125275 * 0^(n-k); where (A125275 * 0^(n-k)) = an infinite lower triangular matrix with A125275: (1, 1, 2, 6, 23, 106, 567, 3434,...) as the main diagonal and the rest zeros.

Corrected definition: Eigentriangle, row sums = A125273, shifted. - Gary W. Adamson, Nov 05 2008

A155862 A 'Morgan Voyce' transform of A007854.

Original entry on oeis.org

1, 4, 22, 130, 790, 4870, 30274, 189202, 1186702, 7461982, 47007034, 296527162, 1872479350, 11833642006, 74833075570, 473463268642, 2996771766046, 18974162475598, 120167557286314, 761214481604554, 4822871486667526, 30561172252753030, 193682023673424226, 1227594333811376050, 7781431761074125486

Offset: 0

Paul Barry, Jan 29 2009

Hankel transform is 3^n*2^binomial(n+1, 2).

Image of A007854 by Riordan array (1/(1-x), x/(1-x)^2).

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

Cf. A001850, A007854, A085478.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(3*Sqrt(1-6*x+x^2) +x -1) )); // G. C. Greubel, Jun 04 2021

CoefficientList[Series[2/(3*Sqrt[1-6*x+x^2]+x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

def A155862_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2/(3*sqrt(1-6*x+x^2) +x-1) ).list()
A155862_list(30) # G. C. Greubel, Jun 04 2021

G.f.: 2/(3*sqrt(1-6*x+x^2) + x - 1).
G.f.: 1/(1 -x -3*x/(1 -x -x/(1 -x -x/(1 -x -x/(1 -x -x/(1- ... (continued fraction).
a(n) = Sum_{k=0..n} binomial(n+k, 2*k)*A007854(k) = Sum_{k=0..n} A085478(n,k) * A007854(k).
2*n*a(n) +(18-25*n)*a(n-1) + 41*(2*n-3)*a(n-2) +(57-25*n)*a(n-3) +2*(n-3)*a(n-4) =0. - R. J. Mathar, Nov 14 2011
a(n) ~ (1+3/sqrt(17)) * (13+3*sqrt(17))^n / 2^(2*n+2). - Vaclav Kotesovec, Feb 01 2014

A155866 A 'Morgan Voyce' transform of the Bell numbers A000110.

Original entry on oeis.org

1, 2, 6, 22, 91, 413, 2032, 10754, 60832, 365815, 2327835, 15612872, 109992442, 811500784, 6253327841, 50211976959, 419239644142, 3632891419054, 32616077413970, 302915722319509, 2906047810600157, 28761123170398258, 293296874302640254, 3078390856651377534, 33220524976632438215

Offset: 0

Paul Barry, Jan 29 2009

Image of Bell numbers under Riordan array (1/(1-x), x/(1-x)^2).

G. C. Greubel, Table of n, a(n) for n = 0..550
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Cf. A000110, A085478.

[(&+[Binomial(n+j,2*j)*Bell(j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 10 2021

A155866[n_]:= Sum[Binomial[n+j, 2*j]*BellB[j], {j,0,n}];
Table[A155866[n], {n, 0, 30}] (* G. C. Greubel, Jun 10 2021 *)

def A155866(n): return sum( binomial(n+j, 2*j)*bell_number(j) for j in (0..n) )
[A155866(n) for n in (0..30)] # G. C. Greubel, Jun 10 2021

G.f.: 1/(1 -x -x/(1 -x -x/(1 -x -x/(1 -x -2*x/(1 -x -x/(1 -x -3*x/(1 -x -x/(1 -x -4*x/(1 - ... (continued fraction).
a(n) = Sum_{k=0..n} binomial(n+k,2k)*A000110(k).
a(n) = Sum_{k=0..n} A085478(n,k)*A000110(k). - Philippe Deléham, Jan 31 2009

A155867 A 'Morgan Voyce' transform of the large Schroeder numbers A006318.

Original entry on oeis.org

1, 3, 13, 65, 355, 2061, 12501, 78323, 503033, 3294373, 21916883, 147708777, 1006330457, 6919474163, 47956087733, 334658965641, 2349535729811, 16583609673797, 117608812053277, 837626242775875, 5988634758319665

Offset: 0

Paul Barry, Jan 29 2009

Image of A006318 under the Riordan array (1/(1-x), x/(1-x)^2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Cf. A006318, A085478.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-3*x+x^2 -Sqrt(1-10*x+19*x^2-10*x^3+x^4))/(2*x*(1-x)) )); // G. C. Greubel, Jun 09 2021

A006318[n_]:= 2*Hypergeometric2F1[-n+1, n+2, 2, -1];
A155867[n_]:= Sum[Binomial[n+j, 2*j]*A006318[j], {j,0,n}];
Table[A155867[n], {n, 0, 40}] (* G. C. Greubel, Jun 09 2021 *)

def A155867_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x+x^2 -sqrt(1-10*x+19*x^2-10*x^3+x^4))/(2*x*(1-x)) ).list()
A155867_list(40) # G. C. Greubel, Jun 09 2021

G.f.: (1 - 3*x + x^2 - sqrt(1 - 10*x + 19*x^2 - 10*x^3 + x^4))/(2*x*(1-x)).
G.f.: 1/(1 -x -2*x/(1 -x -x/(1 -x -2*x/(1 -x -x/(1 -x -2*x/(1 -x -x/(1 - ... (continued fraction).
a(n) = Sum_{k=0..n} binomial(n+k,2k)*A006318(k).
a(n) = Sum_{k=0..n} A085478(n,k)*A006318(k). - Philippe Deléham, Jan 31 2009
Conjecture: (n+1)*a(n) + (4-11*n)*a(n-1) + (29*n-43)*a(n-2) +(73-29*n)*a(n-3) + (11*n-40)*a(n-4) + (5-n)*a(n-5) = 0. - R. J. Mathar, Jul 24 2012
The above recurrence follows from the differential equation (4*x^4 - 14*x^3 + 15*x^2 - 7*x + 1)*A(x) - (x^6 - 11*x^5 + 29*x^4 - 29*x^3 + 11*x^2 - x)*A'(x) + x^4 - x^3 + x - 1 = 0 satisfied by the g.f. A(x). - Peter Bala, Sep 15 2024

A171824 Triangle T(n,k)= binomial(n + k,n) + binomial(2*n-k,n) read by rows.

Original entry on oeis.org

2, 3, 3, 7, 6, 7, 21, 14, 14, 21, 71, 40, 30, 40, 71, 253, 132, 77, 77, 132, 253, 925, 469, 238, 168, 238, 469, 925, 3433, 1724, 828, 450, 450, 828, 1724, 3433, 12871, 6444, 3048, 1452, 990, 1452, 3048, 6444, 12871, 48621, 24320, 11495, 5225, 2717, 2717, 5225, 11495, 24320, 48621

Offset: 0

Roger L. Bagula, Dec 19 2009

			Triangle begins as:
       2;
       3,     3;
       7,     6,     7;
      21,    14,    14,    21;
      71,    40,    30,    40,   71;
     253,   132,    77,    77,  132,  253;
     925,   469,   238,   168,  238,  469, 925;
    3433,  1724,   828,   450,  450,  828, 1724,  3433;
   12871,  6444,  3048,  1452,  990, 1452, 3048,  6444, 12871;
   48621, 24320, 11495,  5225, 2717, 2717, 5225, 11495, 24320, 48621;
  184757, 92389, 43824, 19734, 9009, 6006, 9009, 19734, 43824, 92389, 184757;

G. C. Greubel, Table of n, a(n) for n = 0..1325

Row sums are A000984(n+1).

Cf. A001700, A007318, A054142, A085478.

T:= func< n,k | Binomial(n+k,n) + Binomial(2*n-k,n) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 29 2021

T[n_, k_] = Binomial[n+k, k] + Binomial[2*n-k, n-k];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

def T(n, k): return binomial(n+k,n) + binomial(2*n-k,n)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 29 2021

T(n,k) = A046899(n,k) + A092392(n,k).

Sum_{k=0..n} T(n,k) = binomial(2*n+2, n+1) = 2*A001700(n) = A000984(n+1). - G. C. Greubel, Apr 29 2021

Formula and row sums reference added by the Assoc. Editors of the OEIS, Feb 24 2010

A185331 Riordan array ((1-x+x^2)/(1+x^2), x/(1+x^2)).

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 0, 2, -2, -1, 1, -1, 1, 3, -3, -1, 1, 0, -3, 3, 4, -4, -1, 1, 1, -1, -6, 6, 5, -5, -1, 1, 0, 4, -4, -10, 10, 6, -6, -1, 1, -1, 1, 10, -10, -15, 15, 7, -7, -1, 1, 0, -5, 5, 20, -20, -21, 21, 8, -8, -1, 1

Offset: 0

Philippe Deléham, Feb 08 2012

Triangle T(n,k), read by rows, given by (-1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins:
   1;
  -1,  1;
   0, -1,   1;
   1, -1,  -1,   1;
   0,  2,  -2,  -1,   1;
  -1,  1,   3,  -3,  -1,   1;
   0, -3,   3,   4,  -4,  -1,   1;
   1, -1,  -6,   6,   5,  -5,  -1,  1;
   0,  4,  -4, -10,  10,   6,  -6, -1,  1;
  -1,  1,  10, -10, -15,  15,   7, -7, -1,  1;
   0, -5,   5,  20, -20, -21,  21,  8, -8, -1,  1;
   1, -1, -15,  15,  35, -35, -28, 28,  9, -9, -1, 1;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A206474 (unsigned version).

Cf. A007318, A053122, A078812, A085478, A128908, A129818,

CoefficientList[Series[CoefficientList[Series[(1 - x + x^2)/(1 - y*x + x^2), {x, 0, 10}], x], {y, 0, 10}], y] // Flatten (* G. C. Greubel, Jun 27 2017 *)

T(n,k) = T(n-1,k-1) - T(n-2,k), T(0,0) = 1, T(0,1) = -1, T(0,2) = 0.
G.f.: (1-x+x^2)/(1-y*x+x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A184334(n), A163805(n), A000007(n), A028310(n), A025169(n-1), A005320(n) (n>0) for x = -1, 0, 1, 2, 3, 4 respectively.
T(n,n) = 1, T(n+1,n) = -1, T(n+2,n) = -n, T(n+3,n) = n+1, T(n+4,n) = n(n+1)/2 = A000217(n).
T(2n,2k) = (-1)^(n-k) * A128908(n,k), T(2n+1,2k+1) = -T(2n+1,2k) = A129818(n,k), T(2n+2,2k+1) = (-1)*A053122(n,k). - Philippe Deléham, Feb 09 2012

A206474 Riordan array ((1+x-x^2)/(1-x^2), x/(1-x^2)).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 0, 3, 3, 4, 4, 1, 1, 1, 1, 6, 6, 5, 5, 1, 1, 0, 4, 4, 10, 10, 6, 6, 1, 1, 1, 1, 10, 10, 15, 15, 7, 7, 1, 1, 0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1, 1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1

Offset: 0

Philippe Deléham, Feb 08 2012

Triangle T(n,k), read by rows, given by (1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Antidiagonal sums are A158780(n+1).
Row sums are 2*Fibonacci(n) = 2*A000045(n), n>0.

			Triangle begins :
1
1, 1
0, 1, 1
1, 1, 1, 1
0, 2, 2, 1, 1
1, 1, 3, 3, 1, 1
0, 3, 3, 4, 4, 1, 1
1, 1, 6, 6, 5, 5, 1, 1
0, 4, 4, 10, 10, 6, 6, 1, 1
1, 1, 10, 10, 15, 15, 7, 7, 1, 1
0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1
1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1

Cf. A007318, A078812, A085478, A128908,

t[1, 0] = 1; t[2, 0] = 0; t[n_, n_] = 1; t[n_ /; n >= 0, k_ /; k >= 0] /; k <= n := t[n, k] = t[n-1, k-1] + t[n-2, k]; t[n_, k_] = 0; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)

T(2n, 2k) = A128908(n,k), T(2n+1, 2k) = T(2n+1, 2k+1) = A085478(n,k) = Binomial (n+k, 2k), T(2n+2, 2k+1) = A078812(n,k) = Binomial(n+k-1, 2k-1).
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = T(0,1) = 1, T(0,2) = 0.
G.f.: (1+x-x^2)/(1-x*y-x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n* A000129(n) (n>0), A000007(n), A135528(n-1), A055389(n) for x = -2, -1, 0, 1 respectively .

A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 4, 14, 16, 7, 1, 5, 25, 41, 29, 9, 1, 6, 41, 91, 92, 46, 11, 1, 7, 63, 182, 246, 175, 67, 13, 1, 8, 92, 336, 582, 550, 298, 92, 15, 1, 9, 129, 582, 1254, 1507, 1079, 469, 121, 17, 1, 10, 175, 957, 2508, 3718, 3367, 1925, 696, 154

Offset: 0

Philippe Deléham, Jan 24 2014

Triangle T(n,k), read by rows, given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A111282(n+1) = A025169(n-1).
Diagonal sums are A122391(n+1) = A003945(n-1).

			Triangle begins:
  1;
  1,  1;
  2,  3,   1;
  3,  7,   5,   1;
  4, 14,  16,   7,   1;
  5, 25,  41,  29,   9,  1;
  6, 41,  91,  92,  46, 11,  1;
  7, 63, 182, 246, 175, 67, 13, 1;

Cf. Columns: A028310, A004006.
Cf. Diagonals: A000012, A005408, A130883.
Cf. Similar sequences: A078812, A085478, A111125, A128908, A165253, A207606.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare(1+x/(1-x)^2, 8); # Peter Luschny, Mar 06 2022

CoefficientList[#, y] & /@
CoefficientList[
Series[(1 - x + x^2)/(1 - 2*x - x*y + x^2), {x, 0, 12}], x] (* Wouter Meeussen, Jan 25 2014 *)

G.f.: (1 - x + x^2)/(1 - 2*x - x*y + x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or k > n.
The Riordan square (see A321620) of 1 + x/(1 - x)^2. - Peter Luschny, Mar 06 2022

cp's OEIS Frontend

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

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A101919 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at even heights.

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A144250 Eigentriangle, row sums = A125275, shifted.

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A155862 A 'Morgan Voyce' transform of A007854.

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A155866 A 'Morgan Voyce' transform of the Bell numbers A000110.

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A155867 A 'Morgan Voyce' transform of the large Schroeder numbers A006318.

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A171824 Triangle T(n,k)= binomial(n + k,n) + binomial(2*n-k,n) read by rows.

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