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A227357 de Bruijn's S(6,n).

%I A227357 #19 Feb 16 2025 08:33:20
%S A227357 1,62,38466,41312060,56930297410,90519385516812,157933807781230404,
%T A227357 294111627143303836152,574788682882785699423810,
%U A227357 1165869740380160987511514460,2435635082278794046304453801716,5211959633483650233198112526032152,11377217758058088192513643732271022916
%N A227357 de Bruijn's S(6,n).
%C A227357 Generally, S(s,n) is asymptotic to (2*cos(Pi/(2*s)))^(2*n*s+s-1) *2^(2-s)*(Pi*n)^((1-s)/2)*s^(-1/2).
%D A227357 N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapter 4.7, p.72-75.
%H A227357 Vincenzo Librandi, <a href="/A227357/b227357.txt">Table of n, a(n) for n = 0..200</a>
%H A227357 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>
%F A227357 a(n) ~ (1+sqrt(3))^(12*n+5)/(sqrt(3)*2^(6*n+7)*(Pi*n)^(5/2)).
%F A227357 Recurrence: (n-1)*n^5*(2*n - 1)^3*(78037440*n^10 - 1398170800*n^9 + 11197027400*n^8 - 52776179300*n^7 + 162127296682*n^6 - 339174893304*n^5 + 489377694958*n^4 - 480894941069*n^3 + 308044053231*n^2 - 116166558141*n + 19587964597)*a(n) = (n-1)*(1726812472320*n^18 - 37845973351680*n^17 + 383495168176640*n^16 - 2385128962478080*n^15 + 10193794229981856*n^14 - 31763778392601840*n^13 + 74716717106494000*n^12 - 135540917163836728*n^11 + 192070195278504510*n^10 - 214041209444090466*n^9 + 187905640039584992*n^8 - 129585587008626217*n^7 + 69664459655905576*n^6 - 28800662692839270*n^5 + 8959012339689510*n^4 - 2025094914623067*n^3 + 313623932421492*n^2 - 29741972276520*n + 1302044058000)*a(n-1) - (2*n - 3)*(53979745847040*n^18 - 1291015588175040*n^17 + 14334324120939680*n^16 - 98074075137527840*n^15 + 462828677276119232*n^14 - 1597795252577443036*n^13 + 4175964673926667106*n^12 - 8435559969344133552*n^11 + 13328633341117570446*n^10 - 16565740193886252205*n^9 + 16202242092204003209*n^8 - 12416056458421188647*n^7 + 7385327565692140915*n^6 - 3358099721685530886*n^5 + 1140333781667693872*n^4 - 278997802954150098*n^3 + 46356206084424824*n^2 - 4676191704077040*n + 216042816276000)*a(n-2) + 8*(n-2)^3*(2*n - 5)^5*(2*n - 3)*(78037440*n^10 - 617796400*n^9 + 2125175000*n^8 - 4169616100*n^7 + 5152323982*n^6 - 4181430032*n^5 + 2256662768*n^4 - 801756137*n^3 + 180454862*n^2 - 23380182*n + 1331694)*a(n-3). - _Vaclav Kotesovec_, Sep 27 2016
%p A227357 a:= n->add((-1)^(k+n)*binomial(2*n,k)^6, k=0..2*n):
%p A227357 seq(a(n), n=0..15);  # _Alois P. Heinz_, Jul 17 2013
%t A227357 Table[Sum[(-1)^(k+n)*Binomial[2*n,k]^6, {k, 0, 2*n}], {n, 0, 20}]
%Y A227357 Cf. A000984 (s=2), A006480 (s=3), A050983 (s=4), A050984 (s=5).
%K A227357 nonn
%O A227357 0,2
%A A227357 _Vaclav Kotesovec_, Jul 09 2013