A261440 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) for n >= 0.

1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 3, 4, 0, 1, 1, 4, 11, 9, 0, 1, 1, 5, 21, 46, 21, 0, 1, 1, 6, 34, 127, 207, 51, 0, 1, 1, 7, 50, 268, 833, 979, 127, 0, 1, 1, 8, 69, 485, 2299, 5763, 4797, 323, 0, 1, 1, 9, 91, 794, 5130, 20838, 41401, 24138, 835, 0

Offset: 0

Author

Werner Schulte, Aug 18 2015

Keywords

• nonn
• tabl
• easy

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 0 0 0 0 0 0 0 0 ...

1: 1 1 2 4 9 21 51 127 323 835 ...

2: 1 1 3 11 46 207 979 4797 24138 123998 ...

3: 1 1 4 21 127 833 5763 41401 305877 2309385 ...

4: 1 1 5 34 268 2299 20838 ...

5: 1 1 6 50 485 5130 ...

6: 1 1 7 69 794 ...

7: 1 1 8 91 ...

8: 1 1 9 116 ...

9: 1 1 10 144 ...

10: 1 ...

etc.

For row 1 see A001006, for row 2 see A006605, and for row 3 see A255673.

Be careful if you use the formulas for n < 0 (DIV0, signed values)!

Nevertheless, it might be interesting ...

Conjecture: 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) and (1,-1+2*n) for terms of row n,

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

(d) never cross the x-axis.

This is proved for row 1 (A001006) and row 2 (A006605).

Conjecture: 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), and (1,-1+2*n),

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

(d) never cross the x-axis.

This is proved for B(1,1,k) (A001006), and B(1,2,k) (A006605). - Werner Schulte, Aug 30 2015

Examples

			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.
  1;
  1, 1;
  1, 1, 1;
  1, 1, 2,  0;
  1, 1, 3,  4,  0;
  1, 1, 4, 11,  9,  0;
  1, 1, 5, 21, 46, 21, 0;
  etc.
		

Crossrefs

Cf. A001006, A006605, A255673.

Formula

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

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

P(n,x) = (1 + x*P(n,x)^n)^2/(1 + x*P(n,x)^(n-1)), n > 0.

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} (-1)^j*binomial(n*k+m, j)* binomial(2*n*k + 2*m - 2*j, k - j)), if m > 0, and n > 0 (conjectured).

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

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

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

A(n,4) = n*(8*n^2 + 18*n + 1)/3, n >= 0.

A(n,5) = n*(125*n^3 + 350*n^2 + 55*n - 26)/24, n >= 0.

P(n,x) = exp(Sum_{k>=1} 1/(n*k)*(Sum{j=0..k} (-1)^j*binomial(n*k,j)* binomial(2*n*k-2*j,k-j))) for n > 0 (conjectured). - Werner Schulte, Sep 20 2015

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