A218768 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.

1, 2, 21, 555, 27930, 2297295, 280737765, 47773195470, 10803712844925, 3134455177028175, 1135027918156081950, 501797959522466381775, 265999339071854103540825, 166538120746634232882536250, 121585839578169857291258983125, 102384090433785464586295830691875

Offset: 1

Author

Wesley Transue, Nov 05 2012

Keywords

• easy
• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..226

Programs

Mathematica

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 *)

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

Formula

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

Extensions

Offset corrected by Wesley Transue, Dec 23 2019