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

1, 62, 38466, 41312060, 56930297410, 90519385516812, 157933807781230404, 294111627143303836152, 574788682882785699423810, 1165869740380160987511514460, 2435635082278794046304453801716, 5211959633483650233198112526032152, 11377217758058088192513643732271022916

Offset: 0

Author

Vaclav Kotesovec, Jul 09 2013

Keywords

• nonn

Comments

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).

References

• N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapter 4.7, p.72-75.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Binomial Sums

Crossrefs

Cf. A000984 (s=2), A006480 (s=3), A050983 (s=4), A050984 (s=5).

Programs

Maple

a:= n->add((-1)^(k+n)*binomial(2*n,k)^6, k=0..2*n):
seq(a(n), n=0..15);  # Alois P. Heinz, Jul 17 2013

Mathematica

Table[Sum[(-1)^(k+n)*Binomial[2*n,k]^6, {k, 0, 2*n}], {n, 0, 20}]

Formula

a(n) ~ (1+sqrt(3))^(12*n+5)/(sqrt(3)*2^(6*n+7)*(Pi*n)^(5/2)).

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