A189231 -id:A189231 - cp's OEIS frontend

A189911 Row sums of the extended Catalan triangle A189231.

1, 2, 4, 9, 18, 40, 80, 175, 350, 756, 1512, 3234, 6468, 13728, 27456, 57915, 115830, 243100, 486200, 1016158, 2032316, 4232592, 8465184, 17577014, 35154028, 72804200, 145608400, 300874500, 601749000, 1240940160, 2481880320, 5109183315, 10218366630

Offset: 0

Peter Luschny, May 01 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, The lost Catalan numbers.

Cf. A037965, A097070, A056040, A212303.

A189911 := proc(n) local a,b,d; if n = 0 then 1 else
a := GAMMA(n-floor(n/2)); b := GAMMA(floor(n/2+3/2));
d := GAMMA(floor(n/2+1))^2; GAMMA(n+1)*(a*b+d)/(a*b*d) fi end: seq(A189911(n),n=0..32);
A189911 := proc(n) h:=irem(n,2); g:=iquo(n,2); (g+h+1)*binomial(2*g+h,g+h) end; # Peter Luschny, Oct 24 2013

a[n_] := Module[{q, r}, {q, r} = QuotientRemainder[n, 2]; (q+r+1)*Pochhammer[q+1, q+r]/(q+r)!]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jan 09 2014 *)

def A189911():
    r, n = 1, 1
    while True:
        yield r
        h = n//2
        r *= 2 if is_even(n) else (h+2)*(2*h+1)/(h+1)^2
        n += 1
a = A189911(); [next(a) for i in range(16)]  # Peter Luschny, Oct 24 2013

Let a = Gamma(n-floor(n/2)), b = Gamma(floor(n/2+3/2)), d = Gamma( floor(n/2+1))^2, c = Gamma(n+1). Then a(n) = c*(a*b+d)/(a*b*d).
a(n) = A162246(n,n) + A162246(n,n+1) for n > 0.
From Peter Luschny, Oct 24 2013 : (Start)
E.g.f.: (x+1)*(BesselI(0, 2*x)+BesselI(1, 2*x)).
O.g.f.: I*(2*x^2-1)/(2*sqrt(2*x+1)*x*(2*x-1)^(3/2))-1/(2*x).
Recurrence: a(0) = 1; a(n) = a(n-1)*2 if n is even else ([n/2]+2)*(2*[n/2]+1)/([n/2]+1)^2. ([.] the floor brackets.)
a(n) = A056040(n) + A212303(n) = n$*(1+[(n+1)/2]^((-1)^n)), where n$ is the swinging factorial.
a(2*n) = (n+1)*C(2*n, n) (A037965);
a(2*n+1) = (n+2)*C(2*n+1, n+1) (A097070). (End)
Sum_{n>=0} 1/a(n) = 4*Pi/sqrt(3) - Pi^2/3 - 2. - Amiram Eldar, Aug 20 2022
D-finite with recurrence: (n-2)*(n+1)^2*a(n) - (2*(n-2)^2+2*n-12)*a(n-1) - 4*(n+2)*(n-1)^2*a(n-2) = 0. - Georg Fischer, Nov 25 2022

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

A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 40116600, 1163381400, 155117520, 4808643120, 601080390, 19835652870, 2333606220

Offset: 0

Labos Elemer, Jul 25 2000

nonn

a(n) is the number of 'swinging orbitals' which are enumerated by the trinomial n over [floor(n/2), n mod 2, floor(n/2)].
Similar to but different from A001405(n) = binomial(n, floor(n/2)), a(n) = lcm(A001405(n-1), A001405(n)) (for n>0).
A055773(n) divides a(n), A001316(floor(n/2)) divides a(n).
Exactly p consecutive multiples of p follow the least positive multiple of p if p is an odd prime. Compare with the similar property of A100071. - Peter Luschny, Aug 27 2012
a(n) is the number of vertices of the polytope resulting from the intersection of an n-hypercube with the hyperplane perpendicular to and bisecting one of its long diagonals. - Didier Guillet, Jun 11 2018 [Edited by Peter Munn, Dec 06 2022]

			a(10) = 10!/5!^2 = trinomial(10,[5,0,5]);
a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Didier Guillet, On swinging factorials and the lonely runner conjecture (Text in French).
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Orbitals.
Peter Luschny, Swinging Factorial.

Bisections are A000984 and A002457.

Cf. A000142, A001405, A000188, A055772, A056042, A211226, A162246, A189231, A212303.

[(Factorial(n)/(Factorial(Floor(n/2)))^2): n in [0..40]]; // Vincenzo Librandi, Sep 11 2011

SeriesCoeff := proc(s,n) series(s(w,n),w,n+2);
convert(%,polynom); coeff(%,w,n) end;
a1 := proc(n) local k;
2^(n-(n mod 2))*mul(k^((-1)^(k+1)),k=1..n) end:
a2 := proc(n) option remember;
`if`(n=0,1,n^irem(n,2)*(4/n)^irem(n+1,2)*a2(n-1)) end;
a3 := n -> n!/iquo(n,2)!^2;
g4 := z -> BesselI(0,2*z)*(1+z);
a4 := n -> n!*SeriesCoeff(g4,n);
g5 := z -> (1+z/(1-4*z^2))/sqrt(1-4*z^2);
a5 := n -> SeriesCoeff(g5,n);
g6 := (z,n) -> (1+z^2)^n+n*z*(1+z^2)^(n-1);
a6 := n -> SeriesCoeff(g6,n);
a7 := n -> combinat[multinomial](n,floor(n/2),n mod 2,floor(n/2));
h := n -> binomial(n,floor(n/2)); # A001405
a8 := n -> ilcm(h(n-1),h(n));
F := [a1, a2, a3, a4, a5, a6, a7, a8];
for a in F do seq(a(i), i=0..32) od;

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 02 2010 *)
f[n_] := If[OddQ@n, n*Binomial[n - 1, (n - 1)/2], Binomial[n, n/2]]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 10 2010 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; (* or, twice faster: *) sf[n_] := n!/Quotient[n, 2]!^2; Table[sf[n], {n, 0, 32}] (* Jean-François Alcover, Jul 26 2013, updated Feb 11 2015 *)

a(n)=n!/(n\2)!^2 \\ Charles R Greathouse IV, May 02 2011

def A056040():
    r, n = 1, 0
    while True:
        yield r
        n += 1
        r *= 4/n if is_even(n) else n
a = A056040(); [next(a) for i in range(36)]  # Peter Luschny, Oct 24 2013

a(n) = n!/floor(n/2)!^2. [Essentially the original name.]
a(0) = 1, a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1) for n>=1.
E.g.f.: (1+x)*BesselI(0, 2*x). - Vladeta Jovovic, Jan 19 2004
O.g.f.: a(n) = SeriesCoeff_{n}((1+z/(1-4*z^2))/sqrt(1-4*z^2)).
P.g.f.: a(n) = PolyCoeff_{n}((1+z^2)^n+n*z*(1+z^2)^(n-1)).
a(2n+1) = A046212(2n+1) = A100071(2n+1). - M. F. Hasler, Jan 25 2012
a(2*n) = binomial(2*n,n); a(2*n+1) = (2*n+1)*binomial(2*n,n). Central terms of triangle A211226. - Peter Bala, Apr 10 2012
D-finite with recurrence: n*a(n) + (n-2)*a(n-1) + 4*(-2*n+3)*a(n-2) + 4*(-n+1)*a(n-3) + 16*(n-3)*a(n-4) = 0. - Alexander R. Povolotsky, Aug 17 2012
Sum_{n>=0} 1/a(n) = 4/3 + 8*Pi/(9*sqrt(3)). - Alexander R. Povolotsky, Aug 18 2012
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+1)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
Central column of the coefficients of the swinging polynomials A162246. - Peter Luschny, Oct 22 2013
a(n) = Sum_{k=0..n} A189231(n, 2*k). (Cf. A212303 for the odd case.) - Peter Luschny, Oct 30 2013
a(n) = hypergeometric([-n,-n-1,1/2],[-n-2,1],2)*2^(n-1)*(n+2). - Peter Luschny, Sep 22 2014
a(n) = 4^floor(n/2)*hypergeometric([-floor(n/2), (-1)^n/2], [1], 1). - Peter Luschny, May 19 2015
Sum_{n>=0} (-1)^n/a(n) = 4/3 - 4*Pi/(9*sqrt(3)). - Amiram Eldar, Mar 10 2022

Extended and edited by Peter Luschny, Jun 28 2009

A274709 A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 2, 3, 1, 10, 15, 5, 5, 9, 5, 1, 35, 63, 35, 7, 14, 28, 20, 7, 1, 126, 252, 180, 63, 9, 42, 90, 75, 35, 9, 1, 462, 990, 825, 385, 99, 11, 132, 297, 275, 154, 54, 11, 1, 1716, 3861, 3575, 2002, 702, 143, 13, 429, 1001, 1001, 637, 273, 77, 13, 1

Offset: 0

Peter Luschny, Jul 09 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.
Note that (sum row_n) / row_n(0) = 1,1,2,2,3,3,4,4,..., i.e. the swinging factorials are multiples of the extended Catalan numbers A057977 generalizing the fact that the central binomials are multiples of the Catalan numbers.
T(n, k) is a subtriangle of the extended Catalan triangle A189231.

			Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
[ n] [k=0,1,2,...] [row sum]
[ 0] [  1] 1
[ 1] [  1] 1
[ 2] [  1,   1] 2
[ 3] [  3,   3] 6
[ 4] [  2,   3,   1] 6
[ 5] [ 10,  15,   5] 30
[ 6] [  5,   9,   5,   1] 20
[ 7] [ 35,  63,  35,   7] 140
[ 8] [ 14,  28,  20,   7,  1] 70
[ 9] [126, 252, 180,  63,  9] 630
[10] [ 42,  90,  75,  35,  9,  1] 252
[11] [462, 990, 825, 385, 99, 11] 2772
[12] [132, 297, 275, 154, 54, 11, 1] 924
T(6, 2) = 5 because the five orbitals [-1, 1, 1, 1, -1, -1], [1, -1, 1, 1, -1, -1], [1, 1, -1, -1, -1, 1], [1, 1, -1, -1, 1, -1], [1, 1, -1, 1, -1, -1] raise to maximal height of 2 over the central circle.

Peter Luschny, The lost Catalan numbers
Peter Luschny, Orbitals

Cf. A008313, A039599 (even rows), A047072, A056040 (row sums), A057977 (col 0), A063549 (col 0), A112467, A120730, A189230 (odd rows aerated), A189231, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

S := proc(n,k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, S(n-1,k-1)+
modp(n-k,2)*S(n-1,k)+S(n-1,k+1))) end: T := (n,k) -> S(n,2*k);
seq(print(seq(T(n,k), k=0..iquo(n,2))), n=0..12);

from itertools import accumulate
# Brute force counting
def unit_orbitals(n):
    sym_range = [i for i in range(-n+1, n, 2)]
    for c in Combinations(sym_range, n):
        P = Permutations([sgn(v) for v in c])
        for p in P: yield p
def max_orbitals(n):
    if n == 0: return [1]
    S = [0]*((n+2)//2)
    for u in unit_orbitals(n):
        L = list(accumulate(u))
        S[max(L)] += 1
    return S
for n in (0..10): print(max_orbitals(n))

A238762 Triangle read by rows, generalized ballot numbers, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 3, 1, 0, 2, 0, 2, 0, 3, 0, 8, 0, 10, 1, 0, 3, 0, 5, 0, 5, 0, 4, 0, 15, 0, 30, 0, 35, 1, 0, 4, 0, 9, 0, 14, 0, 14, 0, 5, 0, 24, 0, 63, 0, 112, 0, 126, 1, 0, 5, 0, 14, 0, 28, 0, 42, 0, 42, 0, 6, 0, 35, 0, 112, 0, 252, 0, 420, 0, 462

Offset: 0

Peter Luschny, Mar 05 2014

Compare with the definition of the Motzkin triangle A238763.

			[n\k 0  1  2   3  4   5  6   7]
[0]  1,
[1]  0, 1,
[2]  1, 0, 1,
[3]  0, 2, 0,  3,
[4]  1, 0, 2,  0, 2,
[5]  0, 3, 0,  8, 0, 10,
[6]  1, 0, 3,  0, 5,  0, 5,
[7]  0, 4, 0, 15, 0, 30, 0, 35.

D. E. Knuth, TAOCP, Vol. 4a, Section 7.2.1.6, Eq. 22, p. 451.

P. Luschny, The lost Catalan numbers

Cf. A009766, A189231, A238761, A238879.

binom2 := proc(n, k) local h;
   h := n -> (n-((1-(-1)^n)/2))/2;
   n!/(h(n-k)!*h(n+k)!) end:
A238762 := proc(n, k) local a,b,c;
   a := iquo(n+k+2+modp(n,2), 2);
   b := iquo(n-k+2, 2);
   c := modp(n+k+1, 2);
   binom2(a,b)*b*c/a end:
seq(print(seq(A238762(n, k), k=0..n)), n=0..7);
# Alternativ:
ballot := proc(p, q) option remember;
    if p = 0 and q = 0 then return 1 fi;
    if p < 0 or  p > q then return 0 fi;
    ballot(p-2, q) + ballot(p, q-2);
    if type(q, odd) then % + ballot(p-1, q-1) fi;
    % end:

T[n_, k_] := T[n, k] = Which[k == 0 && n == 0, 1, k < 0 || k > n, 0, True, s = T[n, k - 2] + T[n - 2, k]; If[OddQ[n], s += T[n - 1, k - 1]]; s];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 10 2019, adapted from Sage code *)

@CachedFunction
def ballot(p, q):
    if p == 0 and q == 0: return 1
    if p < 0 or p > q: return 0
    S = ballot(p-2, q) + ballot(p, q-2)
    if q % 2 == 1: S += ballot(p-1, q-1)
    return S
for q in range(8): [ballot(p, q) for p in (0..q)]

Definition: T(0, 0) = 1; T(p, q) = 0 if p < 0 or p > q; T(p, q) = T(p-2, q) + (q mod 2) T(p-1, q-1) + T(p, q-2). (The notation is in the style of Knuth, TAOCP 4a (7.2.1.6)).
T(2*k, 2*n) are the classical ballot numbers A009766(n, k).
T(2*k-1, 2*n-1) = A238761(n, k).
T(n,k) = c*A189231(a, b) with a = floor((n + k + (k mod 2))/2), b = floor((n-k)/2) and c = ((n+k+1) mod 2).
T(n, k) = ((n+k+1) mod 2)*((floor(n/2)+floor(k/2) + 1)^(k mod 2)) * (binomial(floor(n/2) + floor(k/2), floor(n/2)) - binomial(floor(n/2) + floor(k/2), floor(n/2) + 1)).
T(n, k) = ((n+k+1) mod 2)*((floor(n/2)+floor(k/2) + 1)^(k mod 2)) * (floor((n-k)/2) + 1)/(floor(n/2) + 1) * binomial(floor(n/2) + floor(k/2), floor(n/2)).
T(n, n) = A057977(n).
T(n, n-2) = A238452(n-1).
Row sums are A238879.

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

A189230 Complementary Catalan triangle read by rows.

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A189911 Row sums of the extended Catalan triangle A189231.

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A238452 Second column of the extended Catalan triangle A189231.

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A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).

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Magma

Maple

Mathematica

PARI

Sage

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Extensions

A274709 A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.

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Maple

Sage

A238762 Triangle read by rows, generalized ballot numbers, 0<=k<=n.

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Maple

Mathematica

Sage

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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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Maple

Mathematica

Sage

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A189230 Complementary Catalan triangle read by rows.

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