A061187 Staircase of coefficients of polynomials used for column g.f.s of triangle A060924.

3, -2, 6, 2, -4, 9, 39, -57, 30, -8, 12, 136, -96, -84, 104, -32, 15, 320, 293, -1260, 1155, -530, 160, -32, 18, 618, 2118, -4242, 890, 2718, -2652, 1088, -192, 21, 1057, 7224, -5037, -19208, 33383, -23793, 9534, -2632, 672

Offset: 0

Author

Wolfdieter Lang, Apr 20 2001

Keywords

• sign
• easy
• tabf

Comments

a(n,m) is coefficient of x^m of polynomial pLo(n+1,x) := (((1+x)+(3-2*x)*sqrt(x))^(n+1) - ((1+x)-(3-2*x)*sqrt(x))^(n+1))/(2*sqrt(x)) of degree n+1+floor(n/2)= A001651(n). pLo(n+1,x)= sum(binomial(n+1,2*j+1)*(1+x)^(n-2*j)*(3-2*x)^(2*j+1)*x^j,j=0..floor(n/2)), n >= 0.

pLo(m+1,x) appears as numerator polynomial of the g.f. for column m >= 0 of the triangle A060924 (even part of bisection of Lucas triangle).

Examples

			{3, -2}; {6, 2, -4}; {9, 39, -57, 30, -8}; ...; pLo(2, x)= 6+2*x-4*x^2= 2*(1+x)*(3-2*x).
		

Crossrefs

A061186 (companion staircase).

Formula

a(n, m)= sum(3*(-9/2)^j*binomial(n+1, 2*j+1)*sum((-3/2)^(k-m)*binomial(n-2*j, k) *binomial(2*j+1, m-k-j), k=max(0, m-3*j-1)..n-2*j), j=0..floor(n/2)), 0<= m <= n+1+floor(n/2); else 0.