A064304 -id:A064304

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A064094

Triangle composed of generalized Catalan numbers.

+10
25

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

The column m sequence (without leading zeros and the first 1) appears in the Derrida et al. 1992 reference as Z_{N}=Y_{N}(N+1), N >=0, for alpha = m, beta = 1 (or alpha = 1, beta = m). In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1.

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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.

B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.

FORMULA

G.f. for column m: (x^m)/(1-x*c(m*x))= (x^m)*((m-1)+m*x*c(m*x))/(m-1+x) with the g.f. c(x) of Catalan numbers A000108.

T(n, m) = Sum_{j=0..n-m-1} ((n-m-j)*binomial(n-m-1+j, j)*(m^j)/(n-m) or T(n, m) = (1/(1-m))^(n-m)*(1 - m*Sum_{j=0..n-m-1} C(j)*(m*(1-m))^j ), for n - m >= 1, T(n, n) = 1, T(n, m) = 0 if n<m; with C(k) = A000108(k) (Catalan).

EXAMPLE

Triangle begins:

1, 1;

1, 1, 1;

1, 2, 1, 1;

1, 5, 3, 1, 1;

1, 14, 13, 4, 1, 1;

1, 42, 67, 25, 5, 1, 1;

1, 132, 381, 190, 41, 6, 1, 1;

1, 429, 2307, 1606, 413, 61, 7, 1, 1;

1, 1430, 14589, 14506, 4641, 766, 85, 8, 1, 1;

MATHEMATICA

T[n_, 0] = 1; T[n_, 1] := CatalanNumber[n - 1]; T[n_, n_] = 1; T[n_, m_] := (1/(1 - m))^(n - m)*(1 - m*Sum[ CatalanNumber[k]*(m*(1 - m))^k, {k, 0, n - m - 1}]); Table[ T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)

PROG

(Magma)

function A064094(n, k)

if k eq 0 or k eq n then return 1;

else return (&+[(n-k-j)*Binomial(n-k-1+j, j)*k^j: j in [0..n-k-1]])/(n-k);

end if;

end function;

[A064094(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2024

(SageMath)

def A064094(n, k):

if (k==0 or k==n): return 1

else: return sum((n-k-j)*binomial(n-k-1+j, j)*k^j for j in range(n-k))//(n-k)

flatten([[A064094(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 27 2024

CROSSREFS

Diagonals: A000012, A000012, A000027, A001844, A064096, A064302, A064303, A064304, A064305.

Columns (without leading zeros): A000012 (k=0), A000108 (k=1), A064062 (k=2), A064063 (k=3), A064087 (k=4), A064088 (k=5), A064089 (k=6), A064090 (k=7), A064091 (k=8), A064092 (k=9), A064093 (k=10).

Cf. A064095 (row sums).

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Sep 13 2001

STATUS

approved

A323206

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

+10
4

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..54.

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.

FORMULA

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.

EXAMPLE

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, 2, 1

1, 3, 5, 1

1, 4, 13, 14, 1

1, 5, 25, 67, 42, 1

1, 6, 41, 190, 381, 132, 1

MAPLE

# 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;

MATHEMATICA

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]]]

PROG

(Sage) # 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.<x> = PowerSeriesRing(ZZ)

for n in (0..6):

f = (1+(n-1)*x)/((1-x)^2)

s = pseudo_reversion(f, true)

print(s.list())

(PARI)

{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())

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Feb 21 2019

STATUS

approved

A064305

Ninth diagonal of triangle A064094.

+10
3

1, 1430, 95235, 1338790, 9137549, 41260086, 142648495, 409186310, 1022586105, 2298558934, 4750427771, 9170347110, 16730290885, 29104970870, 48618847719, 78419396806, 122678791025, 186826162710

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

FORMULA

a(n) = 1 + 7*n + 27*n^2 + 75*n^3 + 165*n^4 + 297*n^5 + 429*n^6 + 429*n^7, compare to row n = 7 of Catalan triangle A009766.

G.f.: (1 + 1422*x + 83823*x^2 + 616894*x^3 + 1013799*x^4 + 412698*x^5 + 33337*x^6 + 186*x^7)/(1 - x)^8.

E.g.f.: exp(x)*(1 + 1429*x + 46188*x^2 + 176229*x^3 + 181170*x^4 + 66792*x^5 + 9438*x^6 + 429*x^7). - Stefano Spezia, Jul 24 2022

MATHEMATICA

CoefficientList[Series[(1 + 1422 x + 83823 x^2 + 616894 x^3 + 1013799 x^4 + 412698 x^5 + 33337 x^6 + 186 x^7)/(1 - x)^8, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 15 2014 *)

CROSSREFS

Cf. A009766, A064094, A064304 (eighth diagonal).

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 13 2001

STATUS

approved

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