OFFSET
1,3
COMMENTS
The polynomials are pe(n,x) := sum(a(n,m)*x^m,m=0..n-1). Companion polynomials are po(n,x) := sum(b(n,m)*x^m,m=0..n-1) with b(n,m) := A089276(n,m).
LINKS
Wolfdieter Lang, First 7 rows, also for A089276
FORMULA
Combined recursion for polynomials pe(n, x) and po(n, x) defined above: pe(n, x)= 4*(2*n-1)*n*(n-1)*x*po(n-1, x) + (1-(2*n-1)*(2*n-2)*x)*pe(n-1, x) and po(n, x) = 2*(pe(n, x) + ((n-1)/2)*(1-2*n*(2*n-1)*x)*po(n-1, x))/(n+1), n >= 2, with po(1, x) = 1 = pe(1,x). (Corrected Wolfdieter Lang, Apr 11 2013)
Rewritten recursion for polynomial po: po(n, x) = (2*(1 - 2*(2*n-1)*(n-1)*x)*pe(n-1, x) + (n-1)*(1 + 6*n*(2*n-1)*x)* po(n-1, x))/(n+1), with pe(n,x) from above. - Wolfdieter Lang, Apr 11 2013
Combined recursion with b(n, m) := A089276(n, m): a(n, m) = a(n-1, m) - 2*(2*n-1)*(n-1)*a(n-1, m-1) + 4*n*(2*n-1)*(n-1)*b(n-1, m-1) and b(n, m) = (-2*n*(2*n-1)*(n-1)*b(n-1, m-1) + (n-1)*b(n-1, m) + 2*a(n, m))/(n+1), with n >= m+1 >= 2 and a(1, 0)= 1 = b(1, 0), else 0.
Rewritten recursion for triangle b: b(n, m) = (6*n*(2*n-1)*(n-1)*b(n-1, m-1) + (n-1)*b(n-1, m) + 2*a(n-1, m) - 4*(2*n-1)*(n-1)*a(n-1, m-1))/(n+1), with a(n, m) from above. - Wolfdieter Lang, Apr 11 2013
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Nov 07 2003
STATUS
approved