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%I M5183 N2251 #38 Oct 27 2023 19:11:58
%S 1,25,421,6105,83029,1100902,14516426,192422979,2579725656,
%T 35098717902,485534447114,6835409506841,97966603326993,
%U 1429401763567226,21226755241285022,320692032888290224,4926576077469905280,76913478420068425515,1219520974164038038455
%N Number of permutations of length n with longest increasing subsequence of length 5.
%D J. M. Hammersley, A few seedings of research, in Proc. Sixth Berkeley Sympos. Math. Stat. and Prob., ed. L. M. le Cam et al., Univ. Calif. Press, 1972, Vol. I, pp. 345-394.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A001456/b001456.txt">Table of n, a(n) for n = 5..700</a> (first 100 terms from Alois P. Heinz)
%H R. M. Baer and P. Brock, <a href="http://dx.doi.org/10.1090/S0025-5718-1968-0228216-8">Natural sorting over permutation spaces</a>, Math. Comp. 22 1968 385-410.
%F Recurrence: (n-5)*(n+3)^3*(n+4)^2*(n+6)^2*(11025*n^8 + 25515*n^7 - 286443*n^6 - 161641*n^5 + 2585080*n^4 - 59048*n^3 - 7819612*n^2 + 146328*n + 7254720)*a(n) = (n+3)^2*(606375*n^14 + 6629175*n^13 - 3194685*n^12 - 243068077*n^11 - 448185134*n^10 + 2897169968*n^9 + 6909605819*n^8 - 18854806947*n^7 - 49141228309*n^6 + 52949408689*n^5 + 157003723774*n^4 - 31022236184*n^3 - 177627829824*n^2 - 22499155440*n + 46832450400)*a(n-1) - (n-1)*(11278575*n^15 + 107036370*n^14 - 128493459*n^13 - 3499379232*n^12 - 3757671198*n^11 + 38759610078*n^10 + 60611718946*n^9 - 233170832954*n^8 - 421914005785*n^7 + 715791177016*n^6 + 1483014906497*n^5 - 861954416990*n^4 - 2293879983512*n^3 + 206528474736*n^2 + 1232273843856*n + 13490305056)*a(n-2) + (n-2)^2*(n-1)*(84286125*n^13 + 498481200*n^12 - 2434626540*n^11 - 13242031168*n^10 + 26565838790*n^9 + 116444106688*n^8 - 166166829480*n^7 - 520627558844*n^6 + 548244457053*n^5 + 1265779705376*n^4 - 798189974324*n^3 - 1219994476884*n^2 + 526747368888*n + 238058922240)*a(n-3) - 2*(n-3)^2*(n-2)^2*(n-1)*(116181450*n^11 + 631786995*n^10 - 3196642374*n^9 - 12497984441*n^8 + 40113159004*n^7 + 67582342915*n^6 - 249420026774*n^5 - 74467478051*n^4 + 592968590146*n^3 - 201054848490*n^2 - 142917171372*n - 573108048)*a(n-4) + 14400*(n-4)^2*(n-3)^3*(n-2)^2*(n-1)*(11025*n^8 + 113715*n^7 + 200862*n^6 - 727084*n^5 - 854995*n^4 + 4446427*n^3 + 2445184*n^2 - 7589778*n + 1695924)*a(n-5). - _Vaclav Kotesovec_, Mar 16 2014
%F a(n) ~ 9 * 5^(2*n+25/2) / (2^9 * Pi^2 * n^12). - _Vaclav Kotesovec_, Mar 16 2014
%Y Column k=5 of A047874.
%K nonn
%O 5,2
%A _N. J. A. Sloane_
%E More terms from _Alois P. Heinz_, Jul 01 2012
%E Name of the sequence clarified by _Vaclav Kotesovec_, Mar 18 2014
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