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%I #22 Aug 11 2018 04:56:00
%S 1,2,58,4400,644020,155357384,56041398784,28299910066112,
%T 19076135772884080,16558710676700081120,17997592513561138205728,
%U 23948993629880321407298816,38303802347672648465676584704,72510806370598644118983905976320,160368191672482402606757066578885120
%N 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.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.
%H Alois P. Heinz, <a href="/A290306/b290306.txt">Table of n, a(n) for n = 0..206</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eulerian_number#Eulerian_numbers_of_the_second_kind">Eulerian numbers of the second kind</a>
%F a(n) = A201637(2n,n) = A288874(2n,n) = <<2n,n>>, with <<.,.>> = second order Eulerian numbers or Eulerian numbers of the second kind.
%F a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.6899458441572699524424834032837129180107588318196320162637478870996171397... and c = 3.5414537300298411499842602111667139605122817390785452902057395704515855797... - _Vaclav Kotesovec_, Aug 11 2018
%e a(1) = 2: 1122, 1221.
%e a(2) = 58: 11224433, 11244332, 11332244, 11332442, 11334422, 11344322, ..., 44112233, 44112332, 44122133, 44122331, 44123321, 44133122.
%p a:= n-> combinat[eulerian2](2*n, n):
%p seq(a(n), n=0..20);
%p # second Maple program:
%p b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p `if`(n=0, 1, (2*n-k-1)*b(n-1, k-1)+(k+1)*b(n-1, k)))
%p end:
%p a:= n-> b(2*n, n):
%p seq(a(n), n=0..20);
%t 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 *)
%t 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 *)
%Y Cf. A008517, A201637, A112007, A163936, A288874.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Jul 26 2017
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