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A115865
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Legendre_P(n,2)*6^n.
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0
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1, 12, 198, 3672, 71766, 1444392, 29623644, 615614256, 12918175974, 273112332552, 5808412280628, 124127223181776, 2663248527920124, 57334738304731536, 1237861064261885688, 26791929483836768352
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Central coefficients of (1+12x+27x^2)^n. In general, Jacobi_P(n,0,0,sqrt(m))(k*sqrt(m))^n=Legendre_P(n,sqrt(m))(k*sqrt(m))^n has g.f. 1/sqrt(1-2*k*m*x+k^2*x^2), e.g.f. exp(k*m*x)Bessel_I(0,k*sqrt(m(m-1))*x) and gives the central coefficients of (1+k*m*x+k^2*(m(m-1)/4)*x^2)^n.
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FORMULA
| G.f.: 1/sqrt(1-24x+36x^2); E.g.f.: exp(12x)Bessel_I(0,3*sqrt(12)x); a(n)=Jacobi_P(n,0,0,sqrt(4))*(3*sqrt(4))^n; a(n)=3^n*A069835(n).
Conjecture: n*a(n) +12*(1-2*n)*a(n-1) +36*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
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CROSSREFS
| Sequence in context: A034671 A066230 A048667 * A159359 A119864 A036240
Adjacent sequences: A115862 A115863 A115864 * A115866 A115867 A115868
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Feb 01 2006
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