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A115864
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Legendre_P(n,2)*4^n.
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0
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1, 8, 88, 1088, 14176, 190208, 2600704, 36030464, 504047104, 7104278528, 100726755328, 1435037302784, 20526579564544, 294599134674944, 4240277467168768, 61183611081064448, 884741809748967424
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Central coefficients of (1+8x+12x^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(1-16x+16x^2); E.g.f.: exp(8x)Bessel_I(0,2*sqrt(12)x); a(n)=Jacobi_P(n,0,0,sqrt(4))*(2*sqrt(4))^n; a(n)=2^n*A069835(n).
Conjecture: n*a(n) +8*(1-2*n)*a(n-1) +16*(n-1)*a(n-2) =0. - R. J. Mathar, Nov 16 2011
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CROSSREFS
| Sequence in context: A002282 A112907 A053380 * A036917 A003497 A051605
Adjacent sequences: A115861 A115862 A115863 * A115865 A115866 A115867
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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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