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A091528
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Sum {k=1 to n} H(k) k! (n-k)! (mod {n+1}), where H(k) is the k-th harmonic number.
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2
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1, 1, 0, 3, 4, 2, 0, 6, 6, 5, 0, 3, 8, 0, 0, 13, 0, 3, 0, 0, 12, 17, 0, 0, 14, 0, 0, 1, 0, 6, 0, 0, 18, 0, 0, 1, 20, 0, 0, 23, 0, 25, 0, 0, 24, 44, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 30, 8, 0, 36, 32, 0, 0, 0, 0, 10, 0, 0, 0, 2, 0, 56, 38, 0, 0, 0, 0, 19, 0, 0, 42, 48, 0, 0, 44, 0, 0, 6, 0, 0, 0, 0, 48, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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FORMULA
| It appears that the sum {k=1 to n} H(k) k! (n-k)! (mod {n+1}) is congruent to n!*h(n) (mod {n+1}) where h(n) = (1/2)H(n/2) for even n and h(n) = H(n) - (1/2)H(floor(n/2)) for odd n.
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MATHEMATICA
| Table[ Mod[ Sum[ HarmonicNumber[k]k!(n - k)!, {k, 1, n}], n + 1], {n, 1, 95}] (* or *) (from Robert G. Wilson v Jan 14 2004)
h[n_] := If[ EvenQ[n], n!(1/2)HarmonicNumber[n/2], n!(HarmonicNumber[n] - (1/2)HarmonicNumber[ Floor[n/2]])]; Table[ Mod[ h[n], n + 1], {n, 1, 95}]
(* or the following for n > 2000 *) h[n_] := Block[{}, n! Sum[1/(2k - If[ EvenQ[n], 0, 1]), {k, 1, Floor[(n + 1)/2]}], n + 1]]
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CROSSREFS
| Cf. A091529, A091530.
Sequence in context: A077451 A019829 A200125 * A096392 A105825 A145425
Adjacent sequences: A091525 A091526 A091527 * A091529 A091530 A091531
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KEYWORD
| nonn
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AUTHOR
| Leroy Quet Jan 08 2004
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EXTENSIONS
| Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 14 2004
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