OFFSET
0,2
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..206
Wikipedia, Eulerian numbers of the second kind
FORMULA
a(n) = A201637(2n,n) = A288874(2n,n) = <<2n,n>>, with <<.,.>> = second order Eulerian numbers or Eulerian numbers of the second kind.
a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.6899458441572699524424834032837129180107588318196320162637478870996171397... and c = 3.5414537300298411499842602111667139605122817390785452902057395704515855797... - Vaclav Kotesovec, Aug 11 2018
EXAMPLE
a(1) = 2: 1122, 1221.
a(2) = 58: 11224433, 11244332, 11332244, 11332442, 11334422, 11344322, ..., 44112233, 44112332, 44122133, 44122331, 44123321, 44133122.
MAPLE
a:= n-> combinat[eulerian2](2*n, n):
seq(a(n), n=0..20);
# second Maple program:
b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
`if`(n=0, 1, (2*n-k-1)*b(n-1, k-1)+(k+1)*b(n-1, k)))
end:
a:= n-> b(2*n, n):
seq(a(n), n=0..20);
MATHEMATICA
b[n_, k_]:=b[n, k]=If[k<0 || k>n, 0, If[n==0, 1, (2*n - k - 1)*b[n - 1, k - 1] + (k + 1)*b[n - 1, k]]]; Table[b[2n, n], {n, 0, 20}] (* Indranil Ghosh, Jul 27 2017, after second Maple program *)
Flatten[{1, Table[Sum[(-1)^(n-k) * Binomial[4*n + 1, n - k] * StirlingS1[2*n + k, k], {k, 1, n}], {n, 1, 15}]}] (* Vaclav Kotesovec, Aug 11 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 26 2017
STATUS
approved