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A046662
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Sum of mistyped version of binomial coefficients.
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8
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1, 2, 7, 52, 749, 17686, 614227, 29354312, 1844279257, 147273109354, 14561325802271, 1745720380045852, 249461639720702917, 41886684733511640062, 8164388189339113521259, 1828191138807263097870256, 466057478369217965809683377, 134193343258948416556377786322
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OFFSET
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0,2
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COMMENTS
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Origin of the name of this sequence: Binomial coefficients are n!/((n-k)!*k!) but if parentheses are omitted in the denominator, the formula might result in n!/(n-k)!*k! = n!*k!/(n-k)! and the sum giving a(n) instead of 2^n. If k! is forgotten altogether, one gets A000522. - Olivier Gérard, Mar 05 2024
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} n!*k!/(n-k)!.
E.g.f.: exp(x)*F(x), with F(x) = Sum_{k>=0} k!*x^k. - Ralf Stephan, Apr 02 2004
a(n) = n^2*a(n - 1) - n*(n - 1)*a(n - 2) + 1. - Vladeta Jovovic, Jul 15 2004
a(k) == a(0) (mod k) for all k (by the inhomogeneous recurrence equation).
More generally, a(n+k) = a(n) (mod k) for all n and k (by an induction argument on n). It follows that for each positive integer k, the sequence a(n) (mod k) is periodic, with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, ... with exact period 10. (End)
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MATHEMATICA
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Table[Sum[(n!k!)/(n-k)!, {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Sep 29 2012 *)
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CROSSREFS
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Cf. A000522 (Total number of ordered k-subsets of [1,n], k=0..n.)
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KEYWORD
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nonn,easy,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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