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 A229464 Binomial transform of (2*n + 1)!. 2
 1, 7, 133, 5419, 383785, 41782831, 6472067437, 1352114646163, 366325440650449, 124893891684358615, 52323557348796456661, 26420766706149889279867, 15824833185409769038803193, 11092546337733020334329204479, 8995627147680234199615065312445 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Companion sequence to A064570. LINKS FORMULA a(n) = Sum_{k = 0..n} binomial(n,k)*(2*k + 1)!. Clearly a(n) is always odd; indeed, for n >= 1, a(n) = 1 + 6*n*b(n-1), where b(n) = [1, 11, 301, 15991, 1392761, ...] is the binomial transform of A051618. a(n) = Integral_{x >= 0} x*(1 + x^2)^n*exp(-x) dx. a(n) = (2*n + 1)*A064570(n) - 2*n*A064570(n-1). Recurrence equation: a(n) = 1 + 2*n*(2*n + 1)*a(n-1) - 2*n*(2*n - 2)*a(n-2) with a(0) = 1 and a(1) = 7. O.g.f.: Sum_{k >= 0} (2*k + 1)!*x^k/(1 - x)^(k + 1) = 1 + 7*x + 133*x^2 + 5419*x^3 + .... a(n) ~ sqrt(Pi) * 2^(2*n + 2) * n^(2*n + 3/2) / exp(2*n). - Vaclav Kotesovec, Oct 30 2017 From Peter Bala, Nov 26 2017: (Start) E.g.f.: exp(x)*Sum_{n >= 0} A000407(n)*x^n. 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, 7, 3, 9, 5, 1, 7, 3, 9, 5, ... with exact period 5. (End) EXAMPLE a(3) = 1*1! + 3*3! + 3*5! + 1*7! = 5419. MATHEMATICA Table[Sum[Binomial[n, k] * (2*k+1)!, {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 30 2017 *) CROSSREFS Cf. A000522, A051618, A064570, A294352, A000407. Sequence in context: A274258 A251577 A082164 * A317216 A119670 A003374 Adjacent sequences:  A229461 A229462 A229463 * A229465 A229466 A229467 KEYWORD nonn,easy AUTHOR Peter Bala, Sep 25 2013 STATUS approved

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Last modified December 3 18:15 EST 2020. Contains 338908 sequences. (Running on oeis4.)