A062746 Coefficient array for certain polynomials N(3; k,x) (rising powers of x).

1, 3, -3, 1, 12, -29, 30, -15, 3, 55, -222, 405, -417, 252, -84, 12, 273, -1575, 4203, -6678, 6846, -4608, 1980, -495, 55, 1428, -10812, 38367, -83244, 121518, -124146, 89595, -44990, 15015, -3003, 273, 7752, -73017, 325164

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• sign
• easy
• tabf

Comments

The g.f. for the sequence of column r=2*k+1, k >= 0, of array A062745(n,r) is N(3; k,x)*(x^(k+1))/(1-x)^(2*k+2) with N(3; k,x) := sum(a(k,p)*x^p,p=0..2*k).

The m=0 column gives: A001764(n+1). The row sums give A000012 (powers of 1) and (unsigned) A062747.

The sequence of step width of this staircase array is [1,2,2,2,...], i.e. the degree of the row polynomials is [0,2,4,6,...]= A005843.

Examples

			{1}; {3,-3,1}; {12,-29,30,-15,3}; ...; N(3; 1,x)= 3-3*x+x^2.
		

Crossrefs

Cf. A062991.

Formula

a(k, p) := [x^p]N(3; k, x) with N(3; k, x)=(N(3; k-1, x)-A001764(k)*(1-x)^(2*k+1))/x, N(3; 0, x) := 1.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(2*n+1, k+1)*binomial(3*n+1, n)/(3*n+1) if k=0, .., (2*n-3); a(n, k)= ((-1)^k)*binomial(2*n+1, k+1)*binomial(3*n+1, n)/(3*n+1) if k=(2*n-2), ..., 2*n; else 0.