OFFSET
0,3
COMMENTS
The row lengths sequence of this array is 3*n+1 = A016777(n).
For the coefficient array of S(n,x)^3 see A219240. The present array is the even part of the bisection of that one.
The row polynomials in powers of x^2 are (S(2*n,x))^3 =
sum(a(n,m)*x^(2*m), m=0..3*n), n >= 0. The o.g.f. for these row polynomials is GS3even(x,z) = ((z+1)^3 + (1+z)*z*x^2*(3*x^2 - 7))/(((z+1)^2-z*x^2)*((z+1)^2-z*x^2*(x^2-3)^2)). This is obtained from the even part of the bisection of the o.g.f. for A219240.
FORMULA
a(n,m) = [x^m] S(2*n,x)^3, n>=0, 0 <= m <= 3*n.
a(n,m) = [x^m]([z^n]GS3even(x,z)) with GS3even(x,z) the o.g.f. for the row polynomials in powers of x^2, given in a comment above.
EXAMPLE
The array a(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9
0: 1
1: -1 3 -3 1
2: 1 -9 30 -45 30 -9 1
3: 1 18 -123 399 -651 588 -308 93 -15 1
...
Row n=4: [1, -30, 345, -1921, 5598, -9540, 10212, -7137, 3303, -1003, 192, -21, 1],
Row n=5: [-1, 45, -780, 6609, -29847, 80718, -141482, 168927, -141636, 84766, -36366, 11091, -2346, 327, -27, 1],
Row n=6: [1, -63, 1533, -18333, 118029, -460815, 1184872, -2118207, 2729922, -2598297, 1854177, -999687, 407472, -124680, 28164, -4553, 498, -33, 1].
Row n=2: S(4,x)^3 = 1 - 9*x^2 + 30*x^4 - 45*x^6 + 30*x^8 - 9*x^10 + 1*x^12.
CROSSREFS
KEYWORD
sign,easy,tabf
AUTHOR
Wolfdieter Lang, Dec 17 2012
STATUS
approved