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A218768
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a(n+2) = (2*n+1)^2*a(n+1) + (2*n+1)*(2*n-1)*a(n) with a(1)=1 and a(2)=2.
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1
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1, 2, 21, 555, 27930, 2297295, 280737765, 47773195470, 10803712844925, 3134455177028175, 1135027918156081950, 501797959522466381775, 265999339071854103540825, 166538120746634232882536250, 121585839578169857291258983125, 102384090433785464586295830691875
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = (e/Pi)*Integral_{z=-infinity..infinity} z^(2*n) K(0, sqrt(1+z^2)), where K(0, x) is the modified Bessel function of the second kind with order 0 (empirical observation).
a(1) = 1, a(2) = 2, a(n) = (2*n-3)^2*a(n-1) + (2*n-3)*(2*n-5)*a(n-2) for n > 2. - Andrew Howroyd, Dec 24 2019
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MATHEMATICA
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Table[DifferenceRoot[Function[{f, k}, {f[k+2]==(2k+1)^2 f[k+1]+(2k+1)(2k-1)f[k], f[1]==1, f[2]==2}]][n], {n, 1, 15}] (* Corrected by Wesley Transue, Dec 23 2019 *)
Nest[Append[#1, (2 #2 + 1)^2*#1[[#2 + 1]] + (2 #2 + 1) (2 #2 - 1) #1[[-2]]] & @@ {#, Length@ # - 1} &, {1, 2}, 14] (* Michael De Vlieger, Dec 24 2019 *)
RecurrenceTable[{a[1]==1, a[2]==2, a[n+2]==(2n+1)^2 a[n+1]+(2n+1)(2n-1) a[n]}, a, {n, 20}] (* Harvey P. Dale, Sep 12 2021 *)
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PROG
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(PARI) seq(n)={my(a=vector(n)); a[1]=1; a[2]=2; for(n=3, #a, a[n]=(2*n-3)^2*a[n-1] + (2*n-3)*(2*n-5)*a[n-2]); a} \\ Andrew Howroyd, Dec 24 2019
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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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EXTENSIONS
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STATUS
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approved
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