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

0,2

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

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

Sequence in context: A341082 A191798 A206531 * A113635 A136095 A181866

Adjacent sequences: A290303 A290304 A290305 * A290307 A290308 A290309

Keyword

nonn

Author

Alois P. Heinz, Jul 26 2017

Status

approved