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Search: id:A143415
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| A143415 |
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Another sequence of Apery-like numbers for the constant 1/e: a(n) = 1/(n+1)!*sum {k = 0..n-1} C(n-1,k)*(2*n-k)!. |
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+0
3
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| 0, 1, 5, 41, 481, 7421, 142601, 3288205, 88577021, 2731868921, 94969529101, 3675200329841, 156725471006105, 7302990263511541, 369216917569411601, 20130327811188977621, 1177435382675193700021, 73546210385434763486705
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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This sequence is a modified version of A143414.
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FORMULA
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a(n) = 1/(n+1)!*sum {k = 0..n-1} C(n-1,k)*(2*n-k)!. a(n) = 1/(n*(n+1))*A143414(n) for n > 0. Recurrence relation: a(0) = 0, a(1) = 1, (n-1)*(n+1)*a(n) - (n-2)*n*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1) for n >= 2. 1/e = 1/2 - 2 * sum {n = 1..inf} (-1)^(n+1)/(n*(n+2)*a(n)*a(n+1)) = 1/2 - 2*[1/(3*1*5) - 1/(8*5*41) + 1/(15*41*481) - 1/(24*481*7421) + ...] . Conjectural congruences: for r >= 0 and prime p, calculation suggests the congruences a(p^r*(p+1)) == a(p^r) (mod p^(r+1)) may hold.
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MAPLE
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with(combinat): a := n -> 1/(n+1)!*add (binomial(n-1, k)*(2*n-k)!, k = 0..n-1): seq(a(n), n = 0..19);
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CROSSREFS
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Cf. A143413, A143414.
Sequence in context: A096364 A049119 A032188 this_sequence A056545 A009755 A000685
Adjacent sequences: A143412 A143413 A143414 this_sequence A143416 A143417 A143418
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KEYWORD
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easy,nonn
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AUTHOR
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Peter Bala (pbala(AT)toucansurf.com), Aug 14 2008
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