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Search: id:A086880
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| A086880 |
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a(n) = floor( sum(k=0, infinity, k^n/(k!)^2 ) ); related to generalized Bell numbers. |
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+0
2
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| 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
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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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); ...
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FORMULA
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sum(k>=0, k^n/(k!)^2) = A000994(n)*BesI(0, 2) + A000995(n)*BesI(1, 2), using Bessel function values BesI(0, 2)=2.2795853023..., BesI(1, 2)=1.5906368546... 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, ...}.
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EXAMPLE
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a(5) = floor(1^5/(1!)^2 + 2^5/(2!)^2 + 3^5/(3!)^2 + 4^5/(4!)^2 +...)
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CROSSREFS
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Cf. A000994, A000995.
Sequence in context: A082594 A051850 A077013 this_sequence A120405 A155004 A034952
Adjacent sequences: A086877 A086878 A086879 this_sequence A086881 A086882 A086883
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2003
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