This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381889 #14 Mar 19 2025 10:16:56
%S A381889 1,4,18,86,428,2192,11468,60986,328532,1788368,9819128,54302712,
%T A381889 302157424,1690193728,9497996152,53588976802,303434431108,
%U A381889 1723578967056,9818195961512,56071829010968,320970950634288,1841213871449152,10582333064327824,60929582362628968,351385363433883472
%N A381889 Expansion of e.g.f.: (BesselI(0, 2*x) + BesselI(1, 2*x))^2*exp(2*x).
%C A381889 Binomial transform of A151093.
%C A381889 For p prime, a(p) - 2 == 0 (mod 2*p).
%F A381889 a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*A005566(k).
%F A381889 a(n) = Sum_{k=0..n} binomial(n, k)*A001405(n-k)*A001700(k).
%F A381889 a(n) = Sum_{k=0..n} binomial(n, k)*A005773(n-k+1)*A005773(k+1). - _Mélika Tebni_, Mar 19 2025
%p A381889 a := n-> add(binomial(n, k)*binomial(n-k, iquo(n-k,2))*binomial(2*k+1,k+1), k = 0 .. n): seq(a(n), n = 0 .. 24);
%t A381889 len := 24; Table[n!,{n, 0, len}] CoefficientList[Series[(BesselI[0, 2x] + BesselI[1, 2x])^2 Exp[2x], {x, 0, len}], x] (* _Peter Luschny_, Mar 19 2025 *)
%o A381889 (Python)
%o A381889 from math import comb as C
%o A381889 def a(n):
%o A381889 return sum(C(n, k)*2**(n-k)*C(k, k//2)*C(k+1, (k+1)//2) for k in range(n+1))
%o A381889 print([a(n) for n in range(25)])
%o A381889 (PARI) my(x='x+O('x^30)); Vec(serlaplace((besseli(0, 2*x) + x*besseli(1, 2*x))^2*exp(2*x))) \\ _Michel Marcus_, Mar 11 2025
%Y A381889 Cf. A001405, A001700, A005566, A151093.
%K A381889 nonn
%O A381889 0,2
%A A381889 _Mélika Tebni_, Mar 09 2025