A062751 Coefficient array for certain polynomials N(4; k,x) (rising powers in x).

1, 4, -6, 4, -1, 22, -80, 139, -140, 84, -28, 4, 140, -851, 2500, -4536, 5516, -4616, 2640, -990, 220, -22, 969, -8420, 35504, -94584, 175564, -237600, 239250, -179960, 100078, -40040, 10920, -1820, 140, 7084, -80776, 448056

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• sign
• easy
• tabf

Comments

The g.f. for the sequence of column r=3*k+j, k >= 0, j=1,2,3, of the staircase array A062750(n,r) is N(4; k,x)*(x^(k+1))/(1-x)^(3*k+1+j) with N(4; k,x) := sum(a(k,p)*x^p,p=0..3*k).

The m=0 column gives: A002293(n+1). The row sums give A000012 (powers of 1) and (unsigned) A062752.

The sequence of step width of this staircase array is [1,3,3,3,...], i.e. the degree of the row polynomials is [0,3,6,9,...]= A008585.

Examples

			{1}; {4,-6,4,-1}; {22,-80,139,-140,84,-28,4}; ...; N(4; 1,x)= 4-6*x+4*x^2-x^3 =(2-x)*(2-2*x+x^2).
		

Formula

a(k, p) := [x^p]N(4; k, x) with N(4; k, x)=(N(4; k-1, x)-A002293(k)*(1-x)^(3*k+1))/x, N(4; 0, x) := 1.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(3*n+1, k+1)*binomial(4*n+1, n)/(4*n+1) if k=0, .., (3*n-4); a(n, k)= ((-1)^k)*binomial(3*n+1, k+1)*binomial(4*n+1, n)/(4*n+1) if k=(3*n-3), ..., 3*n; else 0.