A238762 -id:A238762 - cp's OEIS frontend

A238879 Row sums of the triangle of generalized ballot numbers A238762.

1, 1, 2, 5, 5, 21, 14, 84, 42, 330, 132, 1287, 429, 5005, 1430, 19448, 4862, 75582, 16796, 293930, 58786, 1144066, 208012, 4457400, 742900, 17383860, 2674440, 67863915, 9694845, 265182525, 35357670, 1037158320, 129644790, 4059928950, 477638700, 15905368710

Offset: 0

Peter Luschny, Mar 06 2014

nonn

Cf. A000108, A002054, A238762, A323844.

A238879 := proc(n) option remember;
if n < 2 then 1 else
   if n mod 2 = 0 then 1/(iquo(n,2)+2)
   else (2*n+4)/((n-1)*(n+5)) fi;
   % *(2*n+2)*A238879(n-2)
fi end:
seq(A238879(i), i = 0..30);

def f():
    f, g, b, n = 1, 1, 1, 1
    while True:
        n += 1
        if b == 1:
            yield g
            g *= 2*(n+1)/(n//2+2)
        else:
            yield f
            f *= 4*(n+1)*(n+2)/((n-1)*(n+5))
        b = 1 - b
A238879 = f(); [next(A238879) for n in range(31)]

a(2n) = A000108(n), a(2n+1) = A002054(n) (conjectured). - Ralf Stephan, Mar 14 2014

A238761 Subtriangle of the generalized ballot numbers, T(n,k) = A238762(2k-1,2n-1), 1<=k<=n, read by rows.

Original entry on oeis.org

1, 2, 3, 3, 8, 10, 4, 15, 30, 35, 5, 24, 63, 112, 126, 6, 35, 112, 252, 420, 462, 7, 48, 180, 480, 990, 1584, 1716, 8, 63, 270, 825, 1980, 3861, 6006, 6435, 9, 80, 385, 1320, 3575, 8008, 15015, 22880, 24310, 10, 99, 528, 2002, 6006, 15015, 32032, 58344, 87516, 92378

Offset: 1

Peter Luschny, Mar 05 2014

			[n\k 1   2    3    4    5    6     7 ]
[1]  1,
[2]  2,  3,
[3]  3,  8,  10,
[4]  4, 15,  30,  35,
[5]  5, 24,  63, 112, 126,
[6]  6, 35, 112, 252, 420,  462,
[7]  7, 48, 180, 480, 990, 1584, 1716.

Row sums are A002054.

Cf. A001700, A009766.

binom2 := proc(n, k) local h;
   h := n -> (n-((1-(-1)^n)/2))/2;
   n!/(h(n-k)!*h(n+k)!) end:
A238761 := (n, k) -> binom2(n+k, n-k+1)*(n-k+1)/(n+k):
seq(print(seq(A238761(n, k), k=1..n)), n=1..7);

h[n_] := (n - ((1 - (-1)^n)/2))/2;
binom2[n_, k_] := n!/(h[n-k]! h[n+k]!);
T[n_, k_] := binom2[n+k, n-k+1] (n-k+1)/(n+k);
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2019, from Maple *)

@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
A238761 = lambda n, k: ballot(2*k-1, 2*n-1)
for n in (1..7): [A238761(n, k) for k in (1..n)]

T(n,n) = A001700(n-1).

T(n,n-1) = A162551(n-1).

A323206 A(n, k) = hypergeometric([-k, k+1], [-k-1], n), square array read by ascending antidiagonals for n,k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 14, 1, 1, 5, 25, 67, 42, 1, 1, 6, 41, 190, 381, 132, 1, 1, 7, 61, 413, 1606, 2307, 429, 1, 1, 8, 85, 766, 4641, 14506, 14589, 1430, 1, 1, 9, 113, 1279, 10746, 55797, 137089, 95235, 4862, 1

Offset: 0

Peter Luschny, Feb 21 2019

Conjecture: A(n, k) is odd if and only if n is even or (n is odd and k + 2 = 2^j for some j > 0).

			Array starts:
    [n\k 0  1    2     3       4        5         6           7  ...]
    [0]  1, 1,   1,    1,      1,       1,        1,          1, ... A000012
    [1]  1, 2,   5,   14,     42,     132,      429,       1430, ... A000108
    [2]  1, 3,  13,   67,    381,    2307,    14589,      95235, ... A064062
    [3]  1, 4,  25,  190,   1606,   14506,   137089,    1338790, ... A064063
    [4]  1, 5,  41,  413,   4641,   55797,   702297,    9137549, ... A064087
    [5]  1, 6,  61,  766,  10746,  161376,  2537781,   41260086, ... A064088
    [6]  1, 7,  85, 1279,  21517,  387607,  7312789,  142648495, ... A064089
    [7]  1, 8, 113, 1982,  38886,  817062, 17981769,  409186310, ... A064090
    [8]  1, 9, 145, 2905,  65121, 1563561, 39322929, 1022586105, ... A064091
         A001844 A064096 A064302  A064303   A064304   A064305  diag: A323209
.
Seen as a triangle (by reading ascending antidiagonals):
                               1
                              1, 1
                            1, 2, 1
                           1, 3, 5, 1
                        1, 4, 13, 14, 1
                      1, 5, 25, 67, 42, 1
                   1, 6, 41, 190, 381, 132, 1

J. Abate, W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011).
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687.

Rows: A000108, A064062, A064063, A064087, A064088, A064089, A064090, A064091.
Columns: A001844, A064096, A064302, A064303, A064304, A064305.
Diagonals: A323209 (main), A323208 (sup main), A323217 (sub main).
Sums of antidiagonals: A323207
Cf. A064094, A009766, A238762.

# The function ballot is defined in A238762.
A := (n, k) -> add(ballot(2*j, 2*k)*n^j, j=0..k):
for n from 0 to 6 do seq(A(n, k), k=0..9) od;
# Or by recurrence:
A := proc(n, k) option remember;
if n = 1 then return `if`(k = 0, 1, (4*k + 2)*A(1, k-1)/(k + 2)) fi:
if k < 2 then return [1, n+1][k+1] fi; n*(4*k - 2);
((%*(n - 1) - k - 1)*A(n, k-1) + %*A(n, k-2))/((n - 1)*(k + 1)) end:
for n from 0 to 6 do seq(A(n, k), k=0..9) od;
# Alternative:
Arow := proc(n, len) # Function REVERT is in Sloane's 'Transforms'.
[seq(1 + n*k, k=0..len-1)]; REVERT(%); seq((-1)^k*%[k+1], k=0..len-1) end:
for n from 0 to 8 do Arow(n, 8) od;

A[n_, k_] := Hypergeometric2F1[-k, k + 1, -k - 1, n];
Table[A[n, k], {n, 0, 8}, {k, 0, 8}]
(* Alternative: *)
prev[f_, n_] := InverseSeries[Series[-x f, {x, 0, n}]]/(-x);
f[n_, x_] := (1 + (n - 1) x)/((1 - x)^2);
For[n = 0, n < 9, n++, Print[CoefficientList[prev[f[n, x], 8], x]]]
(* Continued fraction: *)
num[k_, n_] := If[k < 2, 1, If[k == 2, -x, -n x]];
cf[n_, len_] := ContinuedFractionK[num[k, n], 1, {k, len + 2}];
Arow[n_, len_] := Rest[CoefficientList[Series[cf[n, len], {x, 0, len}], x]];
For[n = 0, n < 9, n++, Print[Arow[n, 8]]]

{A(n,k) = polcoeff((1/x)*serreverse(x*((1+(n-1)*(-x))/((1-(-x))^2)+x*O(x^k))), k)}
for(n=0, 8, for(k=0, 8, print1(A(n, k), ", ")); print())

# Valid for n > 0.
def genCatalan(n): return SR(1/(x- x^2*(1 - sqrt(1 - 4*x*n))/(2*x*n)))
for n in (1..8): print(genCatalan(n).series(x).list())
# Alternative:
def pseudo_reversion(g, invsign=false):
    if invsign: g = g.subs(x=-x)
    g = g.shift(1)
    g = g.reverse()
    g = g.shift(-1)
    return g
R. = PowerSeriesRing(ZZ)
for n in (0..6):
    f = (1+(n-1)*x)/((1-x)^2)
    s = pseudo_reversion(f, true)
    print(s.list())

A(n, k) = [x^k] 1/(x - x^2*C(n*x)) if n > 0 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the generating function of the Catalan numbers A000108.
A(n, k) = Sum_{j=0..k} (binomial(2*k-j, k) - binomial(2*k-j, k+1))*n^(k-j).
A(n, k) = Sum_{j=0..k} binomial(k + j, k)*(1 - j/(k + 1))*n^j (cf. A009766).
A(n, k) = 1 + Sum_{j=0..k-1} ((1+j)*binomial(2*k-j, k+1)/(k-j))*n^(k-j).
A(n, k) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^k)/(1+(n-1)*x), n>0.
A(n, k) ~ ((4*n)^k/(Pi^(1/2)*k^(3/2)))*(1+1/(2*n-1))^2.
If we shift the series f with constant term 1 to the right, invert it with respect to composition and shift the result back to the left then we call this the 'pseudo reversion' of f, prev(f). Row n of the array gives the coefficients of the pseudo reversion of f = (1 + (n - 1)*x)/((1 - x)^2) with an additional inversion of sign. Note that f is not revertible. See also the Sage implementation below.
A(n, k) = [x^k] prev((1 + (n - 1)*(-x))/(1 - (-x))^2).
A(n, k) = [x^(k+1)] cf(n, x) where cf(n, x) = K_{i>=1} c(i)/b(i) in the notation of Gauß with b(i) = 1, c(1) = 1, c(2) = -x and c(i) = -n*x for i > 2.
For a recurrence see the Maple section.

A323208 a(n) = hypergeometric([-n - 1, n + 2], [-n - 2], n).

Original entry on oeis.org

1, 5, 67, 1606, 55797, 2537781, 142648495, 9549411950, 741894295369, 65620725560578, 6511108452179611, 716273662860469000, 86527644431076024637, 11387523335268377432565, 1621766490238904658104583, 248507974510512755641561366, 40769019250019155227631614225

Offset: 0

Peter Luschny, Feb 25 2019

nonn

Cf. A323206, A238762.

# The function ballot is defined in A238762.
a := n -> add(ballot(2*j, 2*n+2)*n^j, j=0..n+1):
seq(a(n), n=0..16);

a[n_] := Hypergeometric2F1[-n - 1, n + 2, -n - 2, n];
Table[a[n], {n, 0, 16}]

a(n) = A323206(n, n+1).
a(n) = Sum_{j=0..n+1} (binomial(2*(n+1)-j,n+1)-binomial(2*(n+1)-j,n+2))*n^(n+1-j).
a(n) = Sum_{j=0..n+1} binomial(n+1+j, n+1)*(1 - j/(n+2))*n^j.
a(n) = 1 + Sum_{j=0..n} ((1+j)*binomial(2*(n+1)-j, n+2)/(n+1-j))*n^(n+1-j).
a(n) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^(n+1))/(1+(n-1)*x), n>0.
a(n) ~ (4^(n + 2)*n^(n + 3))/(sqrt(Pi)*(1 - 2*n)^2*(n + 1)^(3/2)).

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

A323207 a(n) = Sum_{k=0..n} hypergeometric([-k, k + 1], [-k - 1], n - k).

Original entry on oeis.org

1, 2, 4, 10, 33, 141, 752, 4825, 36027, 305132, 2879840, 29909421, 338479429, 4139716658, 54339861530, 761150445734, 11322139144239, 178116143657889, 2952831190016238, 51423702126549166, 938126972940647197, 17883424301972473339

Offset: 0

Peter Luschny, Feb 25 2019

nonn

Cf. A323206, A238762.

# The function ballot is defined in A238762.
A323207 := n -> add(add(ballot(2*j, 2*k)*(n-k)^j, j=0..k), k=0..n):
seq(A323207(n), n=0..21);

a[n_] := Sum[Hypergeometric2F1[-k, k + 1, -k - 1, n - k], {k, 0, n}];
Table[a[n], {n, 0, 21}]

a(n) = Sum_{k=0..n} A323206(n-k, k).
a(n) = Sum_{k=0..n} Sum_{j=0..k} A238762(2*j, 2*k)*(n-k)^j.
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} (binomial(2*(n-k)-j, n-k) - binomial(2*(n-k)-j, n-k+1))*k^(n-k-j).

A323217 a(n) = hypergeometric([-n, n + 1], [-n - 1], n + 1).

Original entry on oeis.org

1, 3, 25, 413, 10746, 387607, 17981769, 1022586105, 68964092542, 5384626548491, 477951767068986, 47546350648784341, 5240644323742274500, 634033030117301108127, 83540992651137240168361, 11908866726507685451458545

Offset: 0

Peter Luschny, Feb 25 2019

nonn

Cf. A323206, A238762.

# The function ballot is defined in A238762.
a := n -> add(ballot(2*j, 2*n)*(n+1)^j, j=0..n):
seq(a(n), n=0..16);

a[n_] := Hypergeometric2F1[-n, n + 1, -n - 1, n + 1];
Table[a[n], {n, 0, 16}]

a(n) = A323206(n+1, n).
a(n) = Sum_{j=0..n} (binomial(2*n-j, n) - binomial(2*n-j, n+1))*(n+1)^(n-j).
a(n) = Sum_{j=0..n} binomial(n+j, n)*(1 - j/(n + 1))*(n + 1)^j.
a(n) = 1 + Sum_{j=0..n-1} ((1+j)*binomial(2*n-j, n+1)/(n-j))*(n+1)^(n-j).
a(n) = (1/(2*Pi))*Integral_{x=0..4*(n+1)} (sqrt(x*(4*(n+1)-x))*x^n)/(1+n*x).
a(n) ~ (4^(n+1)*(n+1)^(n+2))/(sqrt(Pi)*(2*n+1)^2*n^(3/2)).

A238763 A Motzkin triangle read by rows, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 2, 0, 4, 1, 0, 5, 0, 9, 0, 3, 0, 12, 0, 21, 1, 0, 9, 0, 30, 0, 51, 0, 4, 0, 25, 0, 76, 0, 127, 1, 0, 14, 0, 69, 0, 196, 0, 323, 0, 5, 0, 44, 0, 189, 0, 512, 0, 835, 1, 0, 20, 0, 133, 0, 518, 0, 1353, 0, 2188, 0, 6, 0, 70, 0, 392, 0, 1422, 0

Offset: 0

Peter Luschny, Mar 05 2014

Similar to A020474 but with a different enumeration.

Compare with the definition of the generalized ballot numbers A238762.

			[n\k 0  1  2   3  4   5   6   7]
[0]  1,
[1]  0, 1,
[2]  1, 0, 2,
[3]  0, 2, 0, 4,
[4]  1, 0, 5, 0, 9,
[5]  0, 3, 0, 12, 0, 21,
[6]  1, 0, 9, 0, 30, 0, 51,
[7]  0, 4, 0, 25, 0, 76, 0, 127.

Cf. A001006, A005043, A064189, A020474, A238762.

@CachedFunction
def T(p, q):
    if p == 0 and q == 0: return 1
    if p < 0 or  p > q: return 0
    return T(p-2, q) + T(p-1, q-1) + T(p, q-2)
[[T(p, q) for p in (0..q)] for q in (0..9)]

Definition: T(0, 0) = 1; T(p, q) = 0 if p < 0 or p > q; T(p, q) = T(p-2, q) + T(p-1, q-1) + T(p, q-2). (The notation is in the style of Knuth, TAOCP 4a (7.2.1.6)).
T(n, n) = A001006(n).
Sum_{0<=k<=n} T(n, k) = A005043(n+2).

A323209 a(n) = hypergeometric([-n, n + 1], [-n - 1], n).

Original entry on oeis.org

1, 2, 13, 190, 4641, 161376, 7312789, 409186310, 27272680705, 2110472708140, 186023930383501, 18401769878685172, 2018938571514794593, 243319689384354960300, 31955654188732155634341, 4542582850906442990797126, 694922224386422689648830465

Offset: 0

Peter Luschny, Feb 25 2019

nonn

Cf. A323206, A238762.

# The function ballot is defined in A238762.
a := n -> add(ballot(2*k, 2*n)*n^k, k=0..n):
seq(a(n), n=0..16);

a[n_] := Hypergeometric2F1[-n, n + 1, -n - 1, n];
Table[a[n], {n, 0, 14}]

a(n) = A323206(n, n).
a(n) = Sum_{j=0..n} (binomial(2*n-j, n) - binomial(2*n-j, n+1))*n^(n-j).
a(n) = Sum_{j=0..n} binomial(n+j, n)*(1 - j/(n + 1))*n^j.
a(n) = 1 + Sum_{j=0..n-1} ((1+j)*binomial(2*n-j, n+1)/(n-j))*n^(n-j).
a(n) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^n)/(1+(n-1)*x), n>0.
a(n) ~ (4^(n + 1)*n^(n + 1/2))/(sqrt(Pi)*(1 - 2*n)^2).

cp's OEIS Frontend

A238879 Row sums of the triangle of generalized ballot numbers A238762.

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A238761 Subtriangle of the generalized ballot numbers, T(n,k) = A238762(2*k-1,2*n-1), 1<=k<=n, read by rows.

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A323206 A(n, k) = hypergeometric([-k, k+1], [-k-1], n), square array read by ascending antidiagonals for n,k >= 0.

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A323208 a(n) = hypergeometric([-n - 1, n + 2], [-n - 2], n).

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A238452 Second column of the extended Catalan triangle A189231.

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A323207 a(n) = Sum_{k=0..n} hypergeometric([-k, k + 1], [-k - 1], n - k).

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A323217 a(n) = hypergeometric([-n, n + 1], [-n - 1], n + 1).

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A238763 A Motzkin triangle read by rows, 0<=k<=n.

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A238761 Subtriangle of the generalized ballot numbers, T(n,k) = A238762(2k-1,2n-1), 1<=k<=n, read by rows.