A064094 -id:A064094 - cp's OEIS frontend

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

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.

A116868 Triangle of numbers, called Y(1,3), related to generalized Catalan numbers A064063(n) = C(3;n).

Original entry on oeis.org

1, 3, 4, 9, 21, 25, 27, 90, 165, 190, 81, 351, 846, 1416, 1606, 243, 1296, 3834, 8082, 12900, 14506, 729, 4617, 16119, 40365, 79065, 122583, 137089, 2187, 16038, 64395, 185490, 422685, 790434, 1201701, 1338790

Offset: 0

Wolfdieter Lang, Mar 24 2006

This triangle Y(1,3) appears in the totally asymmetric exclusion process for the (unphysical) values alpha=1, beta=3. See the Derrida et al. reference given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.

The main diagonal (M=1) gives the generalized Catalan sequence C(3;n+1):= A064063(n+1).

			Triangle begins:
   1;
   3,   4;
   9,  21,  25;
  27,  90, 165,  190;
  81, 351, 846, 1416, 1606;
  ...

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; eqs. (20), (21), p. 672.
Wolfdieter Lang, First 10 rows.

Cf. A064063.

Row sums give A116862.

G.f. m-th diagonal, m>=1: ((3*x*c(3*x))^m)*(2 + 3*x*c(3*x))/(3*x*(2+x)) with c(x) the o.g.f. of A000108 (Catalan).

A115125 A sequence related to Catalan numbers A000108.

Original entry on oeis.org

1, 2, 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688, 34398208, 240787456, 1704034304, 12171673600, 87636049920, 635361361920, 4634400522240, 33985603829760, 250420238745600, 1853109766717440, 13765958267043840, 102618961627054080, 767411365211013120

Offset: 0

Wolfdieter Lang, Jan 13 2006

Essentially identical to A025225.
The convolution of this sequence with the sequence {(-1)^n} is A064062 (see also A062992).
The sequence A064062 appears in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta=1 (or alpha=1, beta=2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
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; eqs. (20), (21), p. 672.

Cf. A000108, A025225, A062992, A064062, A064094.

[1] cat [2^n*Binomial(2*n-2, n-1)/n: n in [1..30]]; // G. C. Greubel, May 03 2018

a:= n-> `if`(n=0, 1, 2^n*binomial(2*n-2, n-1)/n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2022

a[0] = 1; a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 09 2013 *)

a(n)=if(n==0,1,polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2,n)); \\ Joerg Arndt, May 14 2013

a(n) = C(n-1)*2^n, n>=1, a(0):=1, with C(n):=A000108(n) (Catalan).
G.f.: 1 + (2*x)*c(2*x) with c(x):=(1-sqrt(1-4*x))/(2*x), the o.g.f. of Catalan numbers A000108.
a(n) = A025225(n), n>0. - R. J. Mathar, Aug 11 2008
G.f.: (3 - sqrt(1-8*x))/2 = 2 - U(0) where U(k)=1 - 2*x/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 29 2012
G.f.: 2 - 1/Q(0), where Q(k)= 1 + (8*k+2)*x/(k+1 - x*(2*k+2)*(8*k+6)/(2*x*(8*k+6) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013

A064307 Triangle of coefficients of certain numerator polynomials N(n,x).

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 10, 17, 2, 1, 37, 181, 111, 6, 1, 126, 1530, 2624, 741, 18, 1, 422, 11607, 43940, 34063, 4950, 57, 1, 1422, 83823, 616894, 1013799, 412698, 33337, 186, 1, 4853, 593203, 7846573, 23794925

Offset: 1

Wolfdieter Lang, Sep 13 2001

The g.f. for the sequence in the subdiagonal d>=1 (main diagonal: d=0) of triangle A064094 is N(d,x)/(1-x)^d.

Row sums give A001761(n+1). Main diagonal gives A000957(n+1), n >= 0.

			Triangle begins:
  1;
  1, 0;
  1, 2,  1;       N(3,x) = 1+2*x+x^2 = (1+x)^2.
  1, 10, 17,  2;
  1, 37, 181, 111, 6;
  ...

Cf. A000957, A001761, A064094.

a(n, m) = [x^m]N(n, x); N(n, x) = (1-x)^(n-1) + Sum_{k=1..n-1} A064308(n-1, k)*k!*x^k*(1-x)^(n-1-k) for n >= 2; N(1, x) = 1 = N(2, x).

cp's OEIS Frontend

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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A116868 Triangle of numbers, called Y(1,3), related to generalized Catalan numbers A064063(n) = C(3;n).

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A115125 A sequence related to Catalan numbers A000108.

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A064307 Triangle of coefficients of certain numerator polynomials N(n,x).

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