A086880 a(n) = floor( sum(k=0, infinity, k^n/(k!)^2 ) ); related to generalized Bell numbers.

2, 1, 2, 3, 7, 17, 45, 128, 391, 1287, 4524, 16889, 66657, 276982, 1207598, 5507362, 26203307, 129757596, 667358910, 3558097578, 19632277761, 111930731957, 658482495614, 3992062349412, 24911272290567, 159833355923362

Offset: 0

Author

Paul D. Hanna, Sep 16 2003

Keywords

• nonn

Comments

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); ...

Examples

			a(5) = floor(1^5/(1!)^2 + 2^5/(2!)^2 + 3^5/(3!)^2 + 4^5/(4!)^2 +...)
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..646

Crossrefs

Cf. A000994, A000995, A006789.

Programs

Mathematica

Table[Floor[Sum[k^n/(k!)^2,{k,0,Infinity}]],{n,0,20}] (* Vaclav Kotesovec, Jul 31 2014 *)
Flatten[{2, 1, Table[Floor[HypergeometricPFQ[ConstantArray[2, n-2], ConstantArray[1, n-1], 1]], {n,2,20}]}] (* Vaclav Kotesovec, May 23 2015 *)

Formula

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, ...}.