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 A000313 Number of permutations of length n with 3 consecutive ascending pairs. (Formerly M3633 N1477) 10
 0, 0, 0, 1, 4, 30, 220, 1855, 17304, 177996, 2002440, 24474285, 323060540, 4581585866, 69487385604, 1122488536715, 19242660629360, 348933579412440, 6673354706262864, 134252194678935321, 2834212998777523380, 62651024183503148470, 1447238658638922729580 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Temporary remark: there may be some issues with respect to the offset of this sequence in the formula and program sections. - Joerg Arndt, Nov 16 2014 REFERENCES F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Todd Silvestri, Table of n, a(n) for n = 1..450 (first 100 terms from T. D. Noe) FORMULA a(n) = (n*(n+1)!/6)*sum((-1)^k/k!, k=0..n). a(n) = A065087(n+2)/3. - Zerinvary Lajos, May 25 2007 E.g.f.: x^3/3!*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003 a(n) = round( (exp(-1)*(n+1)!+(-1)^n)*n/6 ). - Mark van Hoeij, Oct 25 2011 G.f.: hypergeom([2, 4],[],x/(x+1))/(x+1)^4. - Mark van Hoeij, Nov 07 2011 a(1) = 0, a(n) = (n-2)*(n-1)*(!(n-2))/6 = (n-2)*(n-1)*A000166(n-2)/6, for n >= 2. - Todd Silvestri, Nov 15 2014 a(n) = hypergeom([4-n,2],[],1)*(-1)^n*A000292(n-3). - Peter Luschny, Nov 19 2014 D-finite with recurrence (-n+4)*a(n) +(n-1)*(n-4)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 01 2022 MAPLE series(hypergeom([2, 4], [], x/(x+1))/(x+1)^4, x=0, 30); # Mark van Hoeij, Nov 07 2011 a := n -> simplify(hypergeom([4-n, 2], [], 1))*(-1)^n*(n-1)*(n-2)*(n-3)/6: seq(a(n), n=1..23); # Peter Luschny, Nov 19 2014 MATHEMATICA Table[(n*(n + 1)!/6)*Sum[(-1)^k/k!, {k, 0, n}], {n, -1, 25}] (* T. D. Noe, Jun 19 2012 *) a[1]:=0; a[n_Integer/; n>=2]:=(n-2) (n-1) Subfactorial[n-2]/6 (* Todd Silvestri, Nov 15 2014 *) PROG (Sage) a = lambda n: (n-2)*(n-1)*sloane.A000166(n-2)/6 if n>2 else 0 [a(n) for n in range(1, 24)] # Peter Luschny, Nov 19 2014 CROSSREFS Cf. A010027, A000255, A000166, A000274, A001260, A001261. A diagonal in triangle A010027. Sequence in context: A134093 A007905 A084976 * A082144 A349456 A220727 Adjacent sequences: A000310 A000311 A000312 * A000314 A000315 A000316 KEYWORD nonn,easy,changed AUTHOR N. J. A. Sloane EXTENSIONS More terms from Vladeta Jovovic, Jan 03 2003 Formula added by Sean A. Irvine, Nov 11 2010 Name clarified and offset changed by N. J. A. Sloane, Apr 12 2014 STATUS approved

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Last modified August 11 07:13 EDT 2024. Contains 375059 sequences. (Running on oeis4.)