A089275 Coefficient triangle of polynomials used for numerator of g.f.s for column sequences of array A078739.

1, 1, 18, 1, 118, 600, 1, 412, 11772, 35280, 1, 1060, 97308, 1494576, 3265920, 1, 2270, 508708, 23753736, 249815520, 439084800, 1, 4298, 1989148, 218417400, 6710001408, 54187574400, 80951270400, 1, 7448, 6355048, 1402502400

Offset: 1

Author

Wolfdieter Lang, Nov 07 2003

Keywords

• nonn
• easy
• tabl

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

Crossrefs

Cf. A078739, 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