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A098269
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a(n) = 2^n*P_n(4), 2^n times the Legendre polynomial of order n at 4.
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4
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1, 8, 94, 1232, 16966, 240368, 3468844, 50712992, 748553926, 11131168688, 166498969924, 2502416381792, 37759888297756, 571681667171168, 8679980422677784, 132116085646644032, 2015249400937940806
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OFFSET
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0,2
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COMMENTS
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Central coefficients of (1+8x+15x^2)^n. 2^n*LegendreP(n,k) yields the central coefficients of (1+2kx+(k^2-1)x^2)^n, with g.f. 1/sqrt(1-4kx+4x^2).
16th binomial transform of 2^n*LegendreP(n,-4) = (-1)^n*A098269(n). - Paul Barry, Sep 03 2004
Diagonal of rational functions 1/(1 + x + 3*y + x*z - 2*x*y*z), 1/(1 - x + y + 3*x*z - 2*x*y*z), 1/(1 - x - x*y - 3*y*z - 2*x*y*z). - Gheorghe Coserea, Jul 03 2018
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LINKS
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FORMULA
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G.f.: 1/sqrt(1-16x+4x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n, k)*binomial(2(n-k), n)*4^(n-2k).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * 3^k * 5^(n-k). - Paul D. Hanna, Sep 29 2012
D-finite with recurrence: n*a(n) = 8*(2*n-1)*a(n-1) - 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(450+120*sqrt(15))*(8+2*sqrt(15))^n/(30*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
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MATHEMATICA
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Table[SeriesCoefficient[1/Sqrt[1-16*x+4*x^2], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
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PROG
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(PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*3^k*5^(n-k))} \\ Paul D. Hanna, Sep 29 2012
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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