A230341 - OEIS

%I #13 Jul 19 2018 09:50:32

%S 1,1,16,293,5811,133669,3574957,109546009,3788091451,145957544981,

%T 6201593613645,288084015576169,14525808779580645,790129980885855401,

%U 46120599397152203401,2875600728737862162481,190740227037467026439611,13411608375592255857753781

%N Number of permutations of [2n] in which the longest increasing run has length n.

%H Alois P. Heinz, <a href="/A230341/b230341.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = A008304(2*n,n).

%F Recurrence (of order 3): n*(n+2)*(12*n^7 - 101*n^6 + 355*n^5 - 668*n^4 + 631*n^3 - 71*n^2 - 344*n + 174)*a(n) = (n+1)*(48*n^9 - 272*n^8 + 453*n^7 - 10*n^6 - 518*n^5 - 741*n^4 + 4090*n^3 - 5810*n^2 + 3444*n - 720)*a(n-1) - 2*n*(2*n - 3)*(60*n^8 - 277*n^7 + 365*n^6 - 59*n^5 - 549*n^4 + 1619*n^3 - 2101*n^2 + 1228*n - 268)*a(n-2) + 4*(n-1)*(2*n - 5)*(2*n - 3)*(12*n^7 - 17*n^6 + n^5 + 12*n^4 - 91*n^3 + 101*n^2 - 12*n - 12)*a(n-3). - _Vaclav Kotesovec_, Jul 16 2014

%F a(n) ~ 2^(2*n+1/2)* n^(n+1) / exp(n). - _Vaclav Kotesovec_, Jul 16 2014

%p a:= proc(n) option remember; `if`(n<5, [1, 1, 16, 293, 5811][n+1],

%p       (2*(n+1)*(26615475780292426*n^4 +2862121494132556*n^3

%p         -240402559504315639*n^2 +79488454158677567*n

%p         +119546195807549142)*a(n-1)

%p       -n*(406022528821033256*n^4 -1031369150352151615*n^3

%p         +11028208356875758*n^2 -1654923205028490137*n

%p         +3900125789057762682)*a(n-2)

%p       +2*(n-1)*(421508861354067594*n^4 -1543365451253363033*n^3

%p         -602924004257000736*n^2 +6654606478117189961*n

%p         -5221800341103267066)*a(n-3)

%p       -4*(2*n-7)*(n-2)*(26655798868586248*n^3 +401269836638339496*n^2

%p         -2000296275137853913*n +2124466470996744981)*a(n-4)

%p       -8*(n-3)*(n-5)*(2*n-7)*(2*n-9)*(8655617328093650*n

%p         -14323734034655567)*a(n-5)) / (n*(n+2)*(13307737890146213*n^2

%p         -43906954139467620*n +22672341406878775)))

%p     end:

%p seq(a(n), n=0..25);

%t b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]];

%t T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1];

%t a[n_] := T[2n, n];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 19 2018, after _Alois P. Heinz_ *)

%Y A diagonal of A008304.

%Y Cf. A267433.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Oct 16 2013