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A089275 Coefficient triangle of polynomials used for numerator of g.f.s for column sequences of array A078739. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..32.

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

Cf. A078739, A089276.

Sequence in context: A040340 A040341 A111872 * A182052 A223520 A242567

Adjacent sequences:  A089272 A089273 A089274 * A089276 A089277 A089278

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Nov 07 2003

STATUS

approved

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Last modified June 13 06:07 EDT 2021. Contains 344981 sequences. (Running on oeis4.)