A262082 Array of coefficients A(n,k) of the formal power series P(n,x) read by upwards antidiagonals, where P(n,x) = Sum_{k>=0} A(n,k)*x^k = 1 + x*P(n,x)^(1*n) + x^2*P(n,x)^(2*n) + x^3*P(n,x)^(3*n) for n >= 0.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 0, 1, 1, 4, 12, 13, 0, 1, 1, 5, 22, 54, 36, 0, 1, 1, 6, 35, 139, 262, 104, 0, 1, 1, 7, 51, 284, 953, 1337, 309, 0, 1, 1, 8, 70, 505, 2509, 6894, 7072, 939, 0, 1, 1, 9, 92, 818, 5455, 23426, 51796, 38426, 2905, 0, 1, 1, 10

Offset

0,9

Comments

The terms define the array A(n,k):

n\k: 0 1 2 3 4 5 6 7 8 9 10 ...

0: 1 1 1 1 0 0 0 0 0 0 0 ...

1: 1 1 2 5 13 36 104 309 939 2905 ...

2: 1 1 3 12 54 262 1337 7072 38426 ...

3: 1 1 4 22 139 953 6894 51796 400269 ...

4: 1 1 5 35 284 2509 23426 ...

5: 1 1 6 51 505 5455 62336 ...

6: 1 1 7 70 818 ...

7: 1 1 8 92 ...

8: 1 1 9 ...

9: 1 1 10 ...

10: 1 1 ...

11: 1 ...

etc.

For row 1 see A036765, for row 2 see A186241, and for row 3 see A200731.

Conjecture 1: The A(n,k), here n > 0, are the number of lattice paths, if

(a) length of path is k*n (for the k-th term of row n),

(b) allowed steps are (1,-1), (1,-1+n), (1,-1+2*n), and (1,-1+3*n),

(c) you start at (0,0), end at (k*n,0), and

(d) never cross the x-axis.

Conjecture 2: The coefficients B(m,n,k) of the P(n,x)^m (see the formula below), m > 0 and n > 0, are the number of lattice paths, if

(a) length of path is k*n+m-1 (k-th coefficient of P(n,x)^m),

(b) allowed steps are (1,-1), (1,-1+n), (1,-1+2*n), and (1,-1+3*n),

(c) you start at (0,m-1), end at (k*n+m-1,0), and

(d) never cross the x-axis.

Links

Table of n, a(n) for n=0..68.

Formula

A(n,k) = 1/(n*k+1) * Sum_{j=0..k} (-2)^j*binomial(n*k+1,j)* binomial(3*n*k+3-2*j,k-j) for n >= 0, and k >= 0. (conjectured)

A(n,0) = A(n,1) = 1, n >= 0;

A(n,2) = n+1, n >= 0;

A(n,3) = (3*n^2+5*n+2)/2, n >= 0;

A(n,4) = (8*n^3+18*n^2+13*n)/3, n >= 0;

A(n,5) = (125*n^4+350*n^3+355*n^2+34*n)/24, n >= 0.

The g.f. P(n,x) of row n of the array A(n,k) satisfy:

P(n,x) = P(n-1,x*P(n,x)), n > 0;

P(n,x) = P(n-2,x*P(n,x)^2), n > 1;

etc.

P(n,x) = P(0,x*P(n,x)^n), n >= 0.

The coefficients B(m,n,k) of the P(n,x)^m are:

B(m,n,k) = m/(n*k+m) * Sum_{j=0..k} (-2)^j*binomial(n*k+m,j)* binomial(3*n*k+3*m-2*j,k-j) for m > 0, n > 0, and k >= 0. (conjectured)

P(n,x) = exp(Sum_{k>=1} 1/(n*k)*(Sum_{j=0..k} (-2)^j*binomial(n*k,j)* binomial(3*n*k-2*j,k-j))) for n > 0 (conjectured); (see for n=1: A036765, for n=2: A186241, and for n=3: A200731).

P(n,x/(1+x+x^2+x^3)^n) = 1+x+x^2+x^3 for n >= 0. - Werner Schulte, Nov 20 2015

Example

The terms of the array A(n,k) read by upwards antidiagonals define the triangle T(n,m) = A(n-m,m) for 0 <= m <= n, i.e.,
n\m 0  1   2   3    4     5     6     7    8  9 ...
0:  1
1:  1  1
2:  1  1   1
3:  1  1   2   1
4:  1  1   3   5    0
5:  1  1   4  12   13     0
6:  1  1   5  22   54    36     0
7:  1  1   6  35  139   262   104     0
8:  1  1   7  51  284   953  1337   309    0
9:  1  1   8  70  505  2509  6894  7072  939  0
etc. [reformatted by Wolfdieter Lang, Oct 15 2015]

Crossrefs

Cf. A036765, A186241, A200731, A261440.

Sequence in context: A124775 A140075 A099555 * A299045 A124530 A243631

Adjacent sequences: A262079 A262080 A262081 * A262083 A262084 A262085

Keyword

nonn,tabl

Author

Werner Schulte, Sep 10 2015

Status

approved