A061186 Staircase of coefficients of polynomials used for column g.f.s of triangle A060923.

1, 1, 1, 1, 11, -11, 4, 1, 30, -6, -23, 12, 1, 58, 123, -278, 193, -72, 16, 1, 95, 565, -715, -145, 601, -360, 80, 1, 141, 1590, 89, -5226, 6441, -3659, 1260, -336, 64, 1, 196, 3549, 6797, -22099, 12369, 9156, -15791, 9492

Offset: 0

Author

Wolfdieter Lang, Apr 20 2001

Keywords

• sign
• easy
• tabf

Comments

a(n,m) is coefficient of x^m of polynomial pLe(n,x) := (((1+x)+(3-2*x)*sqrt(x))^n + ((1+x)-(3-2*x)*sqrt(x))^n)/2 of degree n+floor(n/2)= A032766(n). pLe(n,x)= sum(binomial(n,2*j)*(1+x)^(n-2*j)*(3-2*x)^(2*j)*x^j,j=0..floor(n/2)), n >= 1; pLe(0,x)=1.

pLe(m+1,x) is the numerator polynomial of the g.f. for column m >= 0 of the triangle A060923 (even part of bisection of Lucas triangle).

Examples

			{1}; {1,1}; {1,11,-11,4}; ...; pLe(2,x)= 1+11*x-11*x^2+4*x^3.
		

Crossrefs

A061187 (companion staircase).

Formula

a(n, m)=sum(((-9/2)^j*binomial(n, 2*j)*sum((-3/2)^(k-m)*binomial(n-2*j, k)*binomial(2*j, m-k-j), k=max(0, (m-3*j))..(n-2*j))), j=0..floor(n/2)), 0<= m <= n+floor(n/2); else 0.