A247495 - OEIS

A247495

Generalized Motzkin numbers: Square array read by descending antidiagonals, T(n, k) = k!*[x^k](exp(n*x)* BesselI_{1}(2*x)/x), n>=0, k>=0.

1, 0, 1, 1, 1, 1, 0, 2, 2, 1, 2, 4, 5, 3, 1, 0, 9, 14, 10, 4, 1, 5, 21, 42, 36, 17, 5, 1, 0, 51, 132, 137, 76, 26, 6, 1, 14, 127, 429, 543, 354, 140, 37, 7, 1, 0, 323, 1430, 2219, 1704, 777, 234, 50, 8, 1, 42, 835, 4862, 9285, 8421, 4425, 1514, 364, 65, 9, 1

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OFFSET

0,8

COMMENTS

This two-dimensional array of numbers can be seen as a generalization of the Motzkin numbers A001006 for two reasons: The case n=1 reduces to the Motzkin numbers and the columns are the values of the Motzkin polynomials M_{k}(x) = sum_{j=0..k} A097610(k,j)*x^j evaluated at the nonnegative integers.

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

FORMULA

T(n,k) = (n*(2*k+1)*T(n,k-1)-(n-2)*(n+2)*(k-1)*T(n,k-2))/(k+2) for k>=2.

T(n,k) = sum_{j=0..floor(k/2)}(n^(k-2*j)*binomial(k,2*j)* binomial(2*j,j)/(j+1).

T(n,k) = n^k*hypergeom([(1-k)/2,-k/2], [2], 4/n^2) for n>0.

T(n,n) = A247496(n).

O.g.f. for row n: (1-n*x-sqrt(((n-2)*x-1)*((n+2)*x-1)))/(2*x^2).

O.g.f. for row n: R(x)/x where R(x) is series reversion of x/(1+n*x+x^2).

E.g.f. for row n: exp(n*x)*hypergeom([],[2],x^2).

O.g.f. for column k: the k-th column consists of the values of the k-th Motzkin polynomial M_{k}(x) evaluated at x = 0,1,2,...; M_{k}(x) = sum_{j=0..k} A097610(k,j)*x^j = sum_{j=0..k} (-1)^j*binomial(k,j)*A001006(j)*(x+1)^(k-j).

O.g.f. for column k: sum_{j=0..k} (-1)^(k+1)*A247497(k,j)/(x-1)^(j+1). - Peter Luschny, Dec 14 2014

O.g.f. for row n: 1/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017

T(n,k) is the coefficient of x^k in the expansion of 1/(k+1) * (1 + n*x + x^2)^(k+1). - Seiichi Manyama, May 07 2019

EXAMPLE

Square array starts:

[n\k][0][1] [2] [3] [4] [5] [6] [7] [8]

[0] 1, 0, 1, 0, 2, 0, 5, 0, 14, ... A126120

[1] 1, 1, 2, 4, 9, 21, 51, 127, 323, ... A001006

[2] 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ... A000108

[3] 1, 3, 10, 36, 137, 543, 2219, 9285, 39587, ... A002212

[4] 1, 4, 17, 76, 354, 1704, 8421, 42508, 218318, ... A005572

[5] 1, 5, 26, 140, 777, 4425, 25755, 152675, 919139, ... A182401

[6] 1, 6, 37, 234, 1514, 9996, 67181, 458562, 3172478, ... A025230

A000012,A001477,A002522,A079908, ...

Triangular array starts:

0, 1,

1, 1, 1,

0, 2, 2, 1,

2, 4, 5, 3, 1,

0, 9, 14, 10, 4, 1,

5, 21, 42, 36, 17, 5, 1,

0, 51, 132, 137, 76, 26, 6, 1.

MAPLE

# RECURRENCE

T := proc(n, k) option remember; if k=0 then 1 elif k=1 then n else

(n*(2*k+1)*T(n, k-1)-(n-2)*(n+2)*(k-1)*T(n, k-2))/(k+2) fi end:

seq(print(seq(T(n, k), k=0..9)), n=0..6);

# OGF (row)

ogf := n -> (1-n*x-sqrt(((n-2)*x-1)*((n+2)*x-1)))/(2*x^2):

seq(print(seq(coeff(series(ogf(n), x, 12), x, k), k=0..9)), n=0..6);

# EGF (row)

egf := n -> exp(n*x)*hypergeom([], [2], x^2):

seq(print(seq(k!*coeff(series(egf(n), x, k+2), x, k), k=0..9)), n=0..6);

# MOTZKIN polynomial (column)

A097610 := proc(n, k) if type(n-k, odd) then 0 else n!/(k!*((n-k)/2)!^2* ((n-k)/2+1)) fi end: M := (k, x) -> add(A097610(k, j)*x^j, j=0..k):

seq(print(seq(M(k, n), n=0..9)), k=0..6);

# OGF (column)

col := proc(n, len) local G; G := A247497_row(n); (-1)^(n+1)* add(G[k+1]/(x-1)^(k+1), k=0..n); seq(coeff(series(%, x, len+1), x, j), j=0..len) end: seq(print(col(n, 8)), n=0..6); # Peter Luschny, Dec 14 2014

MATHEMATICA

T[0, k_] := If[EvenQ[k], CatalanNumber[k/2], 0];

T[n_, k_] := n^k*Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4/n^2];

Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

PROG

(Sage)

def A247495(n, k):

if n==0: return(k//2+1)*factorial(k)/factorial(k//2+1)^2 if is_even(k) else 0

return n^k*hypergeometric([(1-k)/2, -k/2], [2], 4/n^2).simplify()

for n in (0..7): print([A247495(n, k) for k in range(11)])

CROSSREFS

Main diagonal gives A247496.