A189911 -id:A189911 - cp's OEIS frontend

A189231 Extended Catalan triangle read by rows.

1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 2, 8, 3, 4, 1, 10, 5, 15, 4, 5, 1, 5, 30, 9, 24, 5, 6, 1, 35, 14, 63, 14, 35, 6, 7, 1, 14, 112, 28, 112, 20, 48, 7, 8, 1, 126, 42, 252, 48, 180, 27, 63, 8, 9, 1, 42, 420, 90, 480, 75, 270, 35, 80, 9, 10, 1, 462, 132, 990, 165, 825, 110, 385, 44, 99, 10, 11, 1

Offset: 0

Peter Luschny, May 01 2011

Let S(n,k) denote the coefficients of the positive powers of the Laurent polynomials C_n(x) = (x+1/x)^(n-1)*(x-1/x)*(x+1/x+n) (if n>0) and C_0(x) = 0.
Then T(n,k) = S(n+1,k+1) for n>=0, k>=0.
The classical Catalan triangle A053121(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is odd.
The complementary Catalan triangle A189230(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is even.
T(n,0) are the extended Catalan numbers A057977(n).

			The Laurent polynomials:
C(0,x) =                 0
C(1,x) =               x - 1/x
C(2,x) =         x^2 + x - 1/x - 1/x^2
C(3,x) = x^3 + 2 x^2 + x - 1/x - 2/x^2 -1/x^3
Triangle T(n,k) = S(n+1,k+1) starts
[0]   1,
[1]   1,  1,
[2]   1,  2,  1,
[3]   3,  2,  3,  1,
[4]   2,  8,  3,  4,  1,
[5]  10,  5, 15,  4,  5,  1,
[6]   5, 30,  9, 24,  5,  6,  1,
[7]  35, 14, 63, 14, 35,  6,  7, 1,
    [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, A189230.

A189231_poly := (n,x)-> `if`(n=0,0,(x+1/x)^(n-2)*(x-1/x)*(x+1/x+n-1)):
seq(print([n],seq(coeff(expand(A189231_poly(n,x)),x,k),k=1..n)),n=1..9);
A189231 := proc(n,k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, A189231(n-1,k-1)+modp(n-k,2)*A189231(n-1,k)+A189231(n-1,k+1))) end:
seq(print(seq(A189231(n,k),k=0..n)),n=0..9);

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]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013 *)

Recurrence: If k>n or k<0 then T(n,k) = 0 else if n=k then T(n,k) = 1; otherwise T(n,k) = T(n-1,k-1) + ((n-k) mod 2)*T(n-1,k) + T(n-1,k+1).
S(n,k) = (k/n)* A162246(n,k) for n>0 where S(n,k) are the coefficients from the definition provided the triangle A162246 is indexed in Laurent style by the recurrence: if abs(k) > n then A162246(n,k) = 0 else if n = k then A162246(n,k) = 1 and otherwise A162246(n,k) = A162246(n-1,k-1)+ modp(n-k,2) * A162246(n-1,k) + A162246(n-1,k+1).
Row sums: A189911(n) = A162246(n,n) + A162246(n,n+1) for n>0.

A212303 a(n) = n!/([(n-1)/2]!*[(n+1)/2]!) for n>0, a(0)=0, and where [ ] = floor.

Original entry on oeis.org

0, 1, 2, 3, 12, 10, 60, 35, 280, 126, 1260, 462, 5544, 1716, 24024, 6435, 102960, 24310, 437580, 92378, 1847560, 352716, 7759752, 1352078, 32449872, 5200300, 135207800, 20058300, 561632400, 77558760, 2326762800, 300540195, 9617286240, 1166803110, 39671305740

Offset: 0

Peter Luschny, Oct 24 2013

nonn

a(n) + A056040(n) = A189911(n), the row sums of the extended Catalan triangle A189231.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Luschny, The lost Catalan numbers.

Cf. A005430, A001700, A056040, A189911.

A212303 := proc(n) if n mod 2 = 0 then n*binomial(n, iquo(n,2))/2 else binomial(n+1, iquo(n,2)+1)/2 fi end: seq(A212303(i), i=0..36);

a[n_?EvenQ] := n*Binomial[n, n/2]/2; a[n_?OddQ] := Binomial[n+1, Quotient[n, 2]+1]/2; Table[a[n], {n, 0, 36}]  (* Jean-François Alcover, Feb 05 2014 *)
nxt[{n_,a_}]:={n+1,If[OddQ[n],a(n+1),(4a(n+1))/(n(n+2))]}; Join[{0}, Transpose[ NestList[ nxt,{1,1},40]][[2]]] (* Harvey P. Dale, Dec 20 2014 *)

def A212303():
    yield 0
    r, n = 1, 1
    while True:
        yield r
        n += 1
        r *= n if is_even(n) else 4*n/((n-1)*(n+1))
a = A212303(); [next(a) for i in range(36)]

E.g.f.: (1+x)*BesselI(1, 2*x).
O.g.f.: -((4*x^2-1)^(3/2)+I-(4*I)*x^2+(4*I)*x^3)/(2*x*(4*x^2-1)^(3/2)).
Recurrence: a(n) = n if n < 2 else a(n) = a(n-1)*n if n is even else a(n-1)*n*4/((n-1)*(n+1)).
a(2*n) = n*C(2*n, n) (A005430); a(2*n+1) = C(2*n+1,  n+1) (A001700).
a(n) = n$*floor((n+1)/2)^((-1)^n), where n$ is the swinging factorial A056040.
a(n) = Sum_{k=0..n} A189231(n, 2*k+1).
Sum_{n>=1} 1/a(n) = 2/3 + (7/27)*sqrt(3)*Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2/3 + Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022

A263841 Expansion of (1 - 2x - x^2)/(sqrt(1+x)(1-3x)^(3/2)2x) - 1/(2x).

Original entry on oeis.org

1, 3, 9, 28, 87, 271, 843, 2619, 8123, 25153, 77763, 240054, 740017, 2278329, 7006093, 21520872, 66039651, 202462113, 620164491, 1898109900, 5805127269, 17741909157, 54188530641, 165405964227, 504601360389, 1538559689751, 4688812503053, 14282580916834, 43486805133903

Offset: 0

N. J. A. Sloane, Nov 02 2015

nonn

A. Asinowski and G. Rote, Point sets with many non-crossing matchings, arXiv preprint arXiv:1502.04925 [cs.CG], 2015. See Table 1.

Cf. A005773, A132894, A189911.

A263841 := n -> add((k+1)*binomial(n, k)*binomial(n-k, iquo(n-k,2)), k = 0 .. n):
seq(A263841(n), n = 0 .. 28); # Mélika Tebni, Jan 25 2024

CoefficientList[Series[(1-2x-x^2)/(Sqrt[1+x] (1-3x)^(3/2) 2x)-1/(2x),{x,0,30}],x] (* Harvey P. Dale, Aug 21 2017 *)

my(x='x+O('x^40)); Vec((1-2*x-x^2)/(sqrt(1+x)*(1-3*x)^(3/2)*2*x)-1/(2*x)) \\ Altug Alkan, Nov 10 2015

D-finite with recurrence: -(n+1)*(n^2+n-3)*a(n) + 2*(n^3+3*n^2-4*n-3)*a(n-1) + 3*(n-1)*(n^2+3*n-1)*a(n-2) = 0. - R. J. Mathar, Feb 17 2016
From Mélika Tebni, Jan 24 2024: (Start)
a(n) = A005773(n+1) + A132894(n).
E.g.f.: (1+x)*exp(x)*(BesselI(0,2*x) + BesselI(1,2*x)). (End)
From Mélika Tebni, Jan 25 2024: (Start)
a(n) = Sum_{k=0..n} A189911(k)*binomial(n,k).
a(n) = Sum_{k=0..n} (k+1)*binomial(n,k)*binomial(n-k,floor((n-k)/2)). (End)

A275329 a(n) = (2+[n/2])n!/((1+[n/2])[n/2]!^2).

Original entry on oeis.org

2, 2, 3, 9, 8, 40, 25, 175, 84, 756, 294, 3234, 1056, 13728, 3861, 57915, 14300, 243100, 53482, 1016158, 201552, 4232592, 764218, 17577014, 2912168, 72804200, 11143500, 300874500, 42791040, 1240940160, 164812365, 5109183315, 636438060, 21002455980, 2463251010

Offset: 0

Peter Luschny, Sep 10 2016

nonn

Cf. A038665, A056040, A057977, A097070, A189911.

a := n -> (2+iquo(n,2))*n!/((1+iquo(n,2))*iquo(n, 2)!^2):
seq(a(n), n=0..34);

def A275329():
    x, n, k = 1, 1, 2
    while True:
        yield x * k
        if is_odd(n):
            x *= n
        else:
            k += 1
            x = (x<<2)//(n+2)
        n += 1
a = A275329(); print([next(a) for _ in range(37)])

a(n) = A056040(n)*(2+[n/2])/(1+[n/2]).
a(n) = A057977(n)*A008619(n+2).
a(2*n+1) = (n+2)*binomial(2*n+1, n+1) = A189911(2*n+1).
a(2*n-3) = n*binomial(2*n-3, n-1) = A097070(n) for n>=2.
a(2*n+2) = (n+3)*binomial(2*n+2, n+1)/(n+2) = A038665(n).
Sum_{n>=0} 1/a(n) = 16/3 - 40*Pi/(9*sqrt(3)) + 4*Pi^2/9. - Amiram Eldar, Aug 20 2022

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A189231 Extended Catalan triangle read by rows.

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A212303 a(n) = n!/([(n-1)/2]!*[(n+1)/2]!) for n>0, a(0)=0, and where [ ] = floor.

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A263841 Expansion of (1 - 2*x - x^2)/(sqrt(1+x)*(1-3*x)^(3/2)*2*x) - 1/(2*x).

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A275329 a(n) = (2+[n/2])*n!/((1+[n/2])*[n/2]!^2).

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A263841 Expansion of (1 - 2x - x^2)/(sqrt(1+x)(1-3x)^(3/2)2x) - 1/(2x).

A275329 a(n) = (2+[n/2])n!/((1+[n/2])[n/2]!^2).