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

%I A089275 #19 Apr 12 2013 13:11:11
%S A089275 1,1,18,1,118,600,1,412,11772,35280,1,1060,97308,1494576,3265920,1,
%T A089275 2270,508708,23753736,249815520,439084800,1,4298,1989148,218417400,
%U A089275 6710001408,54187574400,80951270400,1,7448,6355048,1402502400
%N A089275 Coefficient triangle of polynomials used for numerator of g.f.s for column sequences of array A078739.
%C A089275 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).
%H A089275 Wolfdieter Lang, <a href="/A089275/a089275.pdf">First 7 rows, also for A089276</a>
%F A089275 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)
%F A089275 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
%F A089275 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.
%F A089275 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
%Y A089275 Cf. A078739, A089276.
%K A089275 nonn,easy,tabl
%O A089275 1,3
%A A089275 _Wolfdieter Lang_, Nov 07 2003