A258398 Number of 2n-length strings of balanced parentheses of exactly 10 different types that are introduced in ascending order.

16796, 3233230, 354660460, 29214542500, 2013190058880, 122762429039250, 6850724997273300, 357603651626578500, 17726205673051976100, 843509478504416874150, 38843740303576863755100, 1741683387026398566250500, 76401095775145069217992560

Offset

10,1

Comments

In general, column k>0 of A253180 is asymptotic to (4*k)^n / (k!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

Links

Alois P. Heinz, Table of n, a(n) for n = 10..600

Formula

Recurrence: (n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 110*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 5280*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 145200*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 2524368*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 28865760*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 218683520*(n-8)*(n-7)*(n-6)*(n-5)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-6) + 1076416000*(n-8)*(n-7)*(n-6)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-7) - 3264915456*(n-8)*(n-7)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-8) + 5441863680*(n-8)*(2*n - 17)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-9) - 3715891200*(2*n - 19)*(2*n - 17)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-10). - Vaclav Kotesovec, Jun 01 2015

a(n) ~ 40^n / (10!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

Maple

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i/((k-i)!*i!), i=0..k):
a:= n-> T(n, 10):
seq(a(n), n=10..25);

Crossrefs

Column k=10 of A253180.

Sequence in context: A244107 A264183 A258497 * A215550 A227600 A147698

Adjacent sequences: A258395 A258396 A258397 * A258399 A258400 A258401

Keyword

nonn

Author

Alois P. Heinz, May 28 2015

Status

approved