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

0,1

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); ...

Links

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

Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials, arXiv:2604.01578 [math.NT], 2026. See p. 3.

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, ...}.

Example

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

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 *)

Crossrefs

Cf. A000994, A000995, A006789.

Sequence in context: A376050 A051850 A077013 * A120405 A252889 A155004

Adjacent sequences: A086877 A086878 A086879 * A086881 A086882 A086883

Keyword

nonn

Author

Paul D. Hanna, Sep 16 2003

Status

approved