A290306 Number of permutations of the multiset {1,1,2,2,...,2n,2n} having exactly n ascents and no number smaller than k between the two occurrences of any number k.

1, 2, 58, 4400, 644020, 155357384, 56041398784, 28299910066112, 19076135772884080, 16558710676700081120, 17997592513561138205728, 23948993629880321407298816, 38303802347672648465676584704, 72510806370598644118983905976320, 160368191672482402606757066578885120

Offset: 0

Author

Alois P. Heinz, Jul 26 2017

Keywords

• nonn

Examples

			a(1) = 2: 1122, 1221.
a(2) = 58: 11224433, 11244332, 11332244, 11332442, 11334422, 11344322, ..., 44112233, 44112332, 44122133, 44122331, 44123321, 44133122.
		

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

Crossrefs

Cf. A008517, A201637, A112007, A163936, A288874.

Programs

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 *)

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