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 A119400 #24 Jun 14 2025 15:06:55
%S A119400 1,2,13,172,3809,126526,5874517,362848088,28744087297,2839192902874,
%T A119400 341922922464701,49297062811573732,8380916229314577313,
%U A119400 1658770724530766046422,378056469777362366873989,98286603829297813268996176,28907477297195536067142301697
%N A119400 a(n) = Sum_{k=0..n} (n!/k!)^2*binomial(n,k).
%H A119400 Vincenzo Librandi, <a href="/A119400/b119400.txt">Table of n, a(n) for n = 0..100</a>
%F A119400 Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0,2*sqrt(x/(1-x)))/(1-x).
%F A119400 Recurrence: a(n) = (3*n^2-3*n+2)*a(n-1)-3*(n-1)^4*a(n-2)+(n-2)^3*(n-1)^3*a(n-3). - _Vaclav Kotesovec_, Sep 30 2012
%F A119400 a(n) ~ 1/sqrt(3)*n^(2*n+2/3)/exp(2*n-3*n^(1/3)). - _Vaclav Kotesovec_, Sep 30 2012
%F A119400 E.g.f.: exp(x) * Sum_{n>=0} x^n/n!^3 = Sum_{n>=0} a(n)*x^n/n!^3. - _Paul D. Hanna_, Nov 27 2012
%F A119400 a(n) = Sum_{k=0..n} k!^2*binomial(n,k)^3. - _Ridouane Oudra_, Jun 14 2025
%t A119400 Table[Sum[(n!/k!)^2*Binomial[n, k], {k, 0, n}], {n, 0, 16}] (* _Stefan Steinerberger_, Jun 17 2007 *)
%o A119400 (PARI) a(n)=n!^3*polcoeff(exp(x+x*O(x^n))*sum(m=0, n, x^m/m!^3), n)
%o A119400 for(n=0,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 27 2012
%Y A119400 Cf. A000522, A002720, A216831.
%K A119400 easy,nonn
%O A119400 0,2
%A A119400 _Vladeta Jovovic_, Jul 25 2006
%E A119400 More terms from _Stefan Steinerberger_, Jun 17 2007