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%I #19 Jun 18 2018 19:58:42
%S 2,1,2,3,7,17,45,128,391,1287,4524,16889,66657,276982,1207598,5507362,
%T 26203307,129757596,667358910,3558097578,19632277761,111930731957,
%U 658482495614,3992062349412,24911272290567,159833355923362
%N a(n) = floor( sum(k=0, infinity, k^n/(k!)^2 ) ); related to generalized Bell numbers.
%C Define B(n) = sum(k=0, infinity, k^n/(k!)^2), then there exists a complex linear relation: B(3) = B(2) + B(1); B(4) = 2*B(3); B(5) = 2*B(4) + B(2); B(6) = 5*B(4) + 3*B(2); B(7) = 7*B(5) + B(3); B(12) = B(11) + 11*B(10); ...
%H Vaclav Kotesovec, <a href="/A086880/b086880.txt">Table of n, a(n) for n = 0..646</a>
%F sum(k>=0, k^n/(k!)^2) = A000994(n)*BesselI(0, 2) + A000995(n)*BesselI(1, 2), using Bessel function values BesselI(0, 2)=2.2795853023..., BesselI(1, 2) = 1.5906368546... (A096789) and where A000994 and A000995 shift 2 places left under binomial transform: A000994={1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, ...} A000995={0, 1, 0, 1, 2, 4, 10, 29, 90, 295, 1030, 3838, ...}.
%e a(5) = floor(1^5/(1!)^2 + 2^5/(2!)^2 + 3^5/(3!)^2 + 4^5/(4!)^2 +...)
%t Table[Floor[Sum[k^n/(k!)^2,{k,0,Infinity}]],{n,0,20}] (* _Vaclav Kotesovec_, Jul 31 2014 *)
%t Flatten[{2, 1, Table[Floor[HypergeometricPFQ[ConstantArray[2, n-2], ConstantArray[1, n-1], 1]], {n,2,20}]}] (* _Vaclav Kotesovec_, May 23 2015 *)
%Y Cf. A000994, A000995, A006789.
%K nonn
%O 0,1
%A _Paul D. Hanna_, Sep 16 2003
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