A057977 -id:A057977 - cp's OEIS frontend

A189230 Complementary Catalan triangle read by rows.

0, 1, 0, 0, 2, 0, 3, 0, 3, 0, 0, 8, 0, 4, 0, 10, 0, 15, 0, 5, 0, 0, 30, 0, 24, 0, 6, 0, 35, 0, 63, 0, 35, 0, 7, 0, 0, 112, 0, 112, 0, 48, 0, 8, 0, 126, 0, 252, 0, 180, 0, 63, 0, 9, 0, 0, 420, 0, 480, 0, 270, 0, 80, 0, 10, 0, 462, 0, 990, 0, 825, 0, 385, 0, 99, 0, 11, 0

Offset: 0

Peter Luschny, May 01 2011

T(n,k) = A189231(n,k)*((n - k) mod 2). For comparison: the classical Catalan triangle is A053121(n,k) = A189231(n,k)*((n-k+1) mod 2).

T(n,0) = A138364(n). Row sums: A100071.

			[0]  0,
[1]  1,  0,
[2]  0,  2,  0,
[3]  3,  0,  3,  0,
[4]  0,  8,  0,  4,  0,
[5] 10,  0, 15,  0,  5, 0,
[6]  0, 30,  0, 24,  0, 6, 0,
[7] 35,  0, 63,  0, 35, 0, 7, 0,
   [0],[1],[2],[3],[4],[5],[6],[7]

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, The lost Catalan numbers

Cf. A053121, A162246, A057977, A189231.

A189230 := (n,k) -> A189231(n,k)*modp(n-k,2):
seq(print(seq(A189230(n,k),k=0..n)),n=0..11);

t[n_, k_] /; (k>n || k<0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + Mod[n-k, 2] t[n-1, k] + t[n-1, k+1];
T[n_, k_] := t[n, k] Mod[n-k, 2];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jun 24 2019 *)

A194589 a(n) = A194588(n) - A005043(n); complementary Riordan numbers.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 34, 92, 265, 751, 2156, 6194, 17874, 51702, 149941, 435749, 1268761, 3700391, 10808548, 31613474, 92577784, 271407896, 796484503, 2339561795, 6877992334, 20236257626, 59581937299, 175546527727, 517538571125, 1526679067331, 4505996000730

Offset: 0

Peter Luschny, Aug 30 2011

nonn

The inverse binomial transform of a(n) is A194590(n).

Peter Luschny, The lost Catalan numbers.

Cf. A005043, A189912, A194588, A100071, A005717.

# First method, describes the derivation:
A056040 := n -> n!/iquo(n,2)!^2:
A057977 := n -> A056040(n)/(iquo(n,2)+1);
A001006 := n -> add(binomial(n,k)*A057977(k)*irem(k+1,2),k=0..n):
A005043 := n -> `if`(n=0,1,A001006(n-1)-A005043(n-1)):
A189912 := n -> add(binomial(n,k)*A057977(k),k=0..n):
A194588 := n -> `if`(n=0,1,A189912(n-1)-A194588(n-1)):
A194589 := n -> A194588(n)-A005043(n):
# Second method, more efficient:
A100071 := n -> A056040(n)*(n/2)^(n-1 mod 2):
A194589 := proc(n) local k;
(n mod 2)+(1/2)*add((-1)^k*binomial(n,k)*A100071(k+1),k=1..n) end:
# Alternatively:
a := n -> `if`(n<3,iquo(n,2),hypergeom([1-n/2,-n,3/2-n/2],[1,2-n],4)): seq(simplify(a(n)), n=0..30); # Peter Luschny, Mar 07 2017

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Mod[n, 2] + (1/2)*Sum[(-1)^k*Binomial[n, k]*2^-Mod[k, 2]*(k+1)^Mod[k, 2]*sf[k+1], {k, 1, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 30 2013, from 2nd method *)
Table[If[n < 3, Quotient[n, 2], HypergeometricPFQ[{1 - n/2, -n, 3/2 - n/2}, {1, 2-n}, 4]], {n,0,30}] (* Peter Luschny, Mar 07 2017 *)

a(n):=sum(binomial(n+2,k)*binomial(n-k,k),k,0,(n)/2); /* Vladimir Kruchinin, Sep 28 2015 */

a(n) = sum(k=0, n/2, binomial(n+2,k)*binomial(n-k,k));
vector(30, n, a(n-3)) \\ Altug Alkan, Sep 28 2015

a(n) = sum_{k=0..n} C(n,k)*A194590(k).
a(n) = (n mod 2)+(1/2)*sum_{k=1..n} (-1)^k*C(n,k)*(k+1)$*((k+1)/2)^(k mod 2). Here n$ denotes the swinging factorial A056040(n).
a(n) = PSUMSIGN([0,0,1,2,6,16,45,..] = PSUMSIGN([0,0,A005717]) where PSUMSIGN is from Sloane's "Transformations of integer sequences". - Peter Luschny, Jan 17 2012
A(x) = B'(x)*(1/x^2-1/(B(x)*x)), where B(x)/x is g.f. of A005043. - Vladimir Kruchinin, Sep 28 2015
a(n) = Sum_{k=0..n/2} C(n+2,k)*C(n-k,k). - Vladimir Kruchinin, Sep 28 2015
a(n) = hypergeom([1-n/2,-n,3/2-n/2],[1,2-n],4) for n>=3. - Peter Luschny, Mar 07 2017
a(n) ~ 3^(n + 1/2) / (8*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 17 2024

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).

A238452 Second column of the extended Catalan triangle A189231.

Original entry on oeis.org

0, 1, 2, 2, 8, 5, 30, 14, 112, 42, 420, 132, 1584, 429, 6006, 1430, 22880, 4862, 87516, 16796, 335920, 58786, 1293292, 208012, 4992288, 742900, 19315400, 2674440, 74884320, 9694845, 290845350, 35357670, 1131445440, 129644790, 4407922860, 477638700, 17194993200

Offset: 0

Peter Luschny, Mar 01 2014

Cf. A000108, A057977, A162551, A189231.

a := proc(n) option remember;
  if n < 3 then return n fi;
  if n mod 2 = 0 then return n*a(n-1) fi;
  h := iquo(n,2); n*a(n-1)/(h*(h+2)) end:
seq(a(n), n=0..36);

t[n_, k_] /; (k > n || k < 0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] =
  t[n - 1, k - 1] + Mod[n - k, 2] t[n - 1, k] + t[n - 1, k + 1];
a[n_] := t[n, 1];
Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Jul 10 2019 *)

def A238452():
    a = 1; n = 2
    yield 0
    while True:
        yield a
        a *= n
        if is_odd(n):
            a /= (n//2*(n//2+2))
        n += 1
a = A238452(); [next(a) for n in range(36)]

Definition: a(n) = binomial(n+1, floor(n/2)+1) / (floor(n/2)+2) if n is odd, and 2*binomial(n, floor(n/2)+1) otherwise.
a(n) = A189231(n, 1).
a(n) = A238762(n+1, n-1).
a(2*n) = A162551(n).
a(2*n+1) = A000108(n+1).
a(n) = A057977(n+1) - A057977(n)*((n+1) mod 2). - Peter Luschny, Aug 07 2016

A240558 a(n) = 2^nn!/((floor(n/2)+1)floor(n/2)!^2).

Original entry on oeis.org

1, 2, 4, 24, 32, 320, 320, 4480, 3584, 64512, 43008, 946176, 540672, 14057472, 7028736, 210862080, 93716480, 3186360320, 1274544128, 48432676864, 17611882496, 739699064832, 246566354944, 11342052327424, 3489862254592, 174493112729600, 49855175065600

Offset: 0

Peter Luschny, Apr 14 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A000984, A056040, A057977, A069723, A099045, A151374, A151403.

A240558 := n -> 2^n*n!/((iquo(n,2)+1)*iquo(n,2)!^2):
seq(A240558(n), n=0..30);

Table[SeriesCoefficient[((I*(2*x*(8*x+1)-1))/Sqrt[16*x^2-1]-2*x+1) /(8*x^2), {x,0,n}], {n,0,22}]

x='x+O('x^50); Vec(round((I*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2)) \\ G. C. Greubel, Apr 05 2017

def A240558():
    x, n = 1, 1
    while True:
        yield x
        m = 2*n if is_odd(n) else 8/(n+2)
        x *= m
        n += 1
a = A240558(); [next(a) for i in range(36)]

O.g.f.: ((i*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2), where i=sqrt(-1).
For a recurrence see the Sage program.
a(n) = 2^n*A057977(n)
a(2*k) = A151403(k) = 2^k*A151374(k) = 4^k*A000108(k).
a(2*k+1) = A099045(k+1) = 2^k*A069723(k+2) = 4^k*A000984(k+1).
From Peter Luschny, Jan 31 2015: (Start)
a(n) = Sum_{k=0..n} A056040(n)*C(n,k)/(floor(n/2)+1).
a(n) = Sum_{k=0..n} n!*C(n,k)/((floor(n/2)+1)*(floor(n/2)!)^2).
a(n) = 2^n*n!*[x^n]((x+1)*hypergeom([],[2],x^2)).
a(n) ~ 2^(n+N)/((n+1)^*sqrt(Pi*(2*N+1))); here  = 1 if n is even, 0 otherwise and N = n++1. (End)
Conjecture: -(n+2)*(n^2-5)*a(n) +8*(-2*n-1)*a(n-1) +16*(n-1)*(n^2+2*n-4)*a(n-2)=0. - R. J. Mathar, Jun 14 2016

A275327 Triangle read by rows, Riordan array (1, (2+(x-1)/(2x^2)(1-sqrt(1-4x^2)))/ sqrt(1-4x^2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 2, 7, 3, 1, 0, 10, 10, 12, 4, 1, 0, 5, 33, 25, 18, 5, 1, 0, 35, 42, 78, 48, 25, 6, 1, 0, 14, 144, 144, 155, 80, 33, 7, 1, 0, 126, 168, 420, 356, 275, 122, 42, 8, 1, 0, 42, 610, 723, 1018, 736, 450, 175, 52, 9, 1

Offset: 0

Peter Luschny, Aug 16 2016

			Table starts:
[n] [k=0,1,2,...] row sum
[0] [1] 1
[1] [0, 1] 1
[2] [0, 1, 1] 2
[3] [0, 3, 2, 1] 6
[4] [0, 2, 7, 3, 1] 13
[5] [0, 10, 10, 12, 4, 1] 37
[6] [0, 5, 33, 25, 18, 5, 1] 87
[7] [0, 35, 42, 78, 48, 25, 6, 1] 235
[8] [0, 14, 144, 144, 155, 80, 33, 7, 1] 578
[9] [0, 126, 168, 420, 356, 275, 122, 42, 8, 1] 1518

Cf. A057977 (column 1), A128899, A275328.

S := proc(n, k) option remember; local ecn:
if n = 0 then return n^k fi;
ecn := n -> n!/(iquo(n,2)!^2)/(iquo(n,2)+1);
add(ecn(i)*S(n-1,k-i), i=1..k-n+1) end:
A275327 := (n, k) -> S(k, n):
seq(seq(A275327(n, k),k=0..n),n=0..8);

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, (2+(#-1)/(2#^2) (1-Sqrt[1-4#^2]))/Sqrt[1-4#^2]&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

# uses[riordan_array from A256893]
s = (2+(x-1)/(2*x^2)*(1-sqrt(1-4*x^2)))/sqrt(1-4*x^2)
riordan_array(1, s, 12)

A276666 a(n) = (n-1)*Catalan(n).

Original entry on oeis.org

-1, 0, 2, 10, 42, 168, 660, 2574, 10010, 38896, 151164, 587860, 2288132, 8914800, 34767720, 135727830, 530365050, 2074316640, 8119857900, 31810737420, 124718287980, 489325340400, 1921133836440, 7547311500300, 29667795388452, 116686713634848, 459183826803800

Offset: 0

Peter Luschny, Sep 12 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

A024483 is a variant of this sequence.

Cf. A000108, A000984, A001622, A051631, A056040, A057977, A232500.

Concatenation([-1], List([1..30], n-> 2*Binomial(2*n-1, n+1))); # G. C. Greubel, Aug 29 2019

[(n-1)*Catalan(n): n in [0..30]]; // Vincenzo Librandi, Sep 13 2016

f := (1-3*x)/(x*sqrt(1-4*x))-1/x:
series(f,x,29): seq(coeff(%,x,n), n=0..26);
A276666 := n -> (n^2-1)*(2*n)!/(n+1)!^2:
seq(A276666(n), n=0..26);

Table[(n - 1) CatalanNumber[n], {n, 0, 30}] (* Vincenzo Librandi, Sep 13 2016 *)

a(n) = if(n==0,-1, 2*binomial(2*n-1, n+1)); \\ G. C. Greubel, Aug 29 2019

A276666 = lambda n: (n - 1) * catalan_number(n)
[A276666(n) for n in range(27)]

a(n) = [x^n] (1-3*x)/(x*sqrt(1-4*x))-1/x.
a(n) = 4^n*(n-1)*hypergeom([3/2, -n], [2], 1).
a(n) = 4^n*(n-1)*JacobiP(n,1,-1/2-n,-1)/(n+1).
a(n) = (2*n)! [x^(2^n)]( BesselI(2,2*x) - (1+1/x)*BesselI(1,2*x) ).
a(n) = binomial(2*n,n) - 2*Catalan(n). (See Geoffrey Critzer's formula in A024483).
a(n) = A056040(2*n) - 2*A057977(2*n).
a(n) = A056040(2*n)*(1-2/(n+1)) = (n^2-1)*(2*n)!/(n+1)!^2.
a(n) = A232500(2*n).
a(n) = a(n-1)*2*(n-1)*(2*n-1)/((n-2)*(n+1)) for n > 2. - Chai Wah Wu, Sep 12 2016
a(n) = A024483(n+1) for n>0. - R. J. Mathar, Sep 13 2016
a(n) = A000984(n+1)-3*A000984(n). - Ezhilarasu Velayutham, Aug 27 2019
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=2} 1/a(n) = 5/6 - Pi/(9*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 26*sqrt(5)*log(phi)/25 - 7/10, where phi is the golden ratio (A001622). (End)

A189913 Triangle read by rows: T(n,k) = binomial(n, k) * k! / (floor(k/2)! * floor((k+2)/2)!).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 3, 1, 4, 6, 12, 2, 1, 5, 10, 30, 10, 10, 1, 6, 15, 60, 30, 60, 5, 1, 7, 21, 105, 70, 210, 35, 35, 1, 8, 28, 168, 140, 560, 140, 280, 14, 1, 9, 36, 252, 252, 1260, 420, 1260, 126, 126, 1, 10, 45, 360, 420, 2520, 1050, 4200, 630, 1260, 42

Offset: 0

Peter Luschny, May 24 2011

The triangle may be regarded a generalization of the triangle A097610:
A097610(n,k) = binomial(n,k)*(2*k)$/(k+1);
T(n,k) = binomial(n,k)*(k)$/(floor(k/2)+1).
Here n$ denotes the swinging factorial A056040(n). As A097610 is a decomposition of the Motzkin numbers A001006, 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, 2,  1
[3]  1, 3,  3,   3
[4]  1, 4,  6,  12,  2
[5]  1, 5, 10,  30, 10,  10
[6]  1, 6, 15,  60, 30,  60,  5
[7]  1, 7, 21, 105, 70, 210, 35, 35

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Peter Luschny, The lost Catalan numbers.

Row sums are A189912.

Cf. A097610, A057977, A001006.

/* As triangle */ [[Binomial(n,k)*Factorial(k)/(Factorial(Floor(k/2))*Factorial(Floor((k + 2)/2))): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 13 2018

A189913 := (n,k) -> binomial(n,k)*(k!/iquo(k,2)!^2)/(iquo(k,2)+1):
seq(print(seq(A189913(n,k),k=0..n)),n=0..7);

T[n_, k_] := Binomial[n, k]*k!/((Floor[k/2])!*(Floor[(k + 2)/2])!); Table[T[n, k], {n, 0, 10}, {k, 0, n}]// Flatten (* G. C. Greubel, Jan 13 2018 *)

{T(n,k) = binomial(n,k)*k!/((floor(k/2))!*(floor((k+2)/2))!) };
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 13 2018

From R. J. Mathar, Jun 07 2011: (Start)
T(n,1) = n.
T(n,2) = A000217(n-1).
T(n,3) = A027480(n-2).
T(n,4) = A034827(n). (End)

A237884 a(n) = (n!m)/(m!(m+1)!) where m = floor(n/2).

Original entry on oeis.org

0, 0, 1, 3, 4, 20, 15, 105, 56, 504, 210, 2310, 792, 10296, 3003, 45045, 11440, 194480, 43758, 831402, 167960, 3527160, 646646, 14872858, 2496144, 62403600, 9657700, 260757900, 37442160, 1085822640, 145422675, 4508102925, 565722720, 18668849760, 2203961430

Offset: 0

Peter Luschny, Feb 14 2014

nonn

A237884 := proc(n) m := iquo(n,2); (n!*m)/(m!*(m+1)!) end;
seq(A237884(n), n = 0..34);

CoefficientList[Series[-((-1 + Sqrt[1 - 4 x^2] -x (-1 + Sqrt[1 - 4 x^2] +
2 x (-3 + 2 Sqrt[1 - 4 x^2] +x (3 + 4 x - 2 Sqrt[1 - 4 x^2]))))/
(2 x^2 (1 - 4 x^2)^(3/2))), {x, 0, 30}], x] (* Benedict W. J. Irwin, Aug 15 2016 *)
Table[(n! #)/(#! (# + 1)!) &@ Floor[n/2], {n, 0, 34}] (* Michael De Vlieger, Aug 15 2016 *)

def A237884():
    r, s, n = 1, 0, 0
    while True:
        yield s
        n += 1
        r *= 4/n if is_even(n) else n
        s = r * (n//2)/(n//2+1)
a = A237884(); [next(a) for i in range(35)]

a(2*n) = A001791(n).
a(2*n+1) = A000917(n-1).
a(n) = n^(n mod 2)*binomial(2*floor(n/2), floor(n/2)-1).
a(n) = A162246(n, n+2) = n!/((n-ceiling((n+2)/2))!*floor((n+2)/2)!) if n > 1, otherwise 0.
a(n) = A056040(n)*floor(n/2)/(floor(n/2)+1).
a(n) + A056040(n) = A057977(n).
G.f.: -((p - 1 - x*(p - 1 + 2*x*(2*p - 3 + x*(3 + 4*x - 2*p))))/(2*x^2*p^3)), where p=sqrt(1-4*x^2). - Benedict W. J. Irwin, Aug 15 2016

A275326 Triangle read by rows, T(n,k) = ceiling(A275325(n,k)/2) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 3, 0, 2, 1, 0, 10, 5, 0, 5, 4, 1, 0, 35, 28, 7, 0, 14, 14, 6, 1, 0, 126, 126, 54, 9, 0, 42, 48, 27, 8, 1, 0, 462, 528, 297, 88, 11, 0, 132, 165, 110, 44, 10, 1, 0, 1716, 2145, 1430, 572, 130, 13, 0, 429, 572, 429, 208, 65, 12, 1

Offset: 0

Peter Luschny, Aug 15 2016

An extension of the Catalan triangle A128899.

			Triangle starts:
[ n] [k=0,1,2,...] [row sum]
[ 0] [1] 1
[ 1] [0, 1] 1
[ 2] [0, 1] 1
[ 3] [0, 3] 3
[ 4] [0, 2, 1] 3
[ 5] [0, 10, 5] 15
[ 6] [0, 5, 4, 1] 10
[ 7] [0, 35, 28, 7] 70
[ 8] [0, 14, 14, 6, 1] 35
[ 9] [0, 126, 126, 54,  9] 315
[10] [0, 42, 48, 27, 8, 1] 126
[11] [0, 462, 528, 297, 88, 11] 1386
[12] [0, 132, 165, 110, 44, 10, 1] 462

Peter Luschny, Orbitals

Cf. A057977, A093178, A128899, A275324 (row sums), A275325.

# uses[orbital_factors]
# Function orbital_factors is in A275325.
def half_orbital_factors(n):
    F = orbital_factors(n)
    return [f//2 for f in F] if n >= 2 else F
for n in (0..12): print(half_orbital_factors(n))

T(n,k) = A275325(n,k)/2 for n>=2.
T(n,1) = A057977(n) for n>=1 (the extended Catalan numbers).
For odd n: T(n,1) = Sum_{k>=0} T(n+1,k).
Main diagonal: T(n, floor(n/2)) = A093178(n).

cp's OEIS Frontend

A189230 Complementary Catalan triangle read by rows.

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Maple

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A194589 a(n) = A194588(n) - A005043(n); complementary Riordan numbers.

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Maxima

PARI

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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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A238452 Second column of the extended Catalan triangle A189231.

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Sage

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A240558 a(n) = 2^n*n!/((floor(n/2)+1)*floor(n/2)!^2).

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PARI

Sage

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A275327 Triangle read by rows, Riordan array (1, (2+(x-1)/(2*x^2)*(1-sqrt(1-4*x^2)))/ sqrt(1-4*x^2)).

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Sage

A276666 a(n) = (n-1)*Catalan(n).

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GAP

Magma

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A240558 a(n) = 2^nn!/((floor(n/2)+1)floor(n/2)!^2).

A275327 Triangle read by rows, Riordan array (1, (2+(x-1)/(2x^2)(1-sqrt(1-4x^2)))/ sqrt(1-4x^2)).

A237884 a(n) = (n!m)/(m!(m+1)!) where m = floor(n/2).