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
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.
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
Comments