%I M5126 N2221 #61 Aug 11 2018 02:55:20

%S 1,22,328,4400,58140,785304,11026296,162186912,2507481216,40788301824,

%T 697929436800,12550904017920,236908271543040,4687098165573120,

%U 97049168010017280,2099830209402931200,47405948832458496000,1115089078488795648000,27290469545695931904000,694002594415741341696000

%N Eulerian numbers of the second kind: <<n+3, n>>.

%C Eulerian permutations of the multiset {1,1,2,2,...,n+3,n+3} with n ascents.Eulerian permutations have the restriction that for all m, all integers between the two copies of m are less than m. In particular, the two 1s are always next to each other.

%C The sequence gives the Eulerian numbers <<3,0>>, <<4,1>>, <<5,2>>, <<6,3>>, ... (and in particular the offset is 0).

%D Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2nd edition; Addison-Wesley, 1994, pp. 270-271.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002539/b002539.txt">Table of n, a(n) for n=0..99</a>

%H L. Carlitz, <a href="/A002538/a002538.pdf">Some numbers related to the Stirling numbers of the first and second kind</a>, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., Numbers 544-576 (1976): 49-55. [Annotated scanned copy. The triangle is A008517.]

%H I. Gessel and R. P. Stanley, <a href="https://doi.org/10.1016/0097-3165(78)90042-0">Stirling polynomials</a>, J. Combin. Theory, A 24 (1978), 24-33.

%H O. J. Munch, <a href="/A000460/a000460.pdf">Om potensproduktsummer</a> [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]

%H O. J. Munch, <a href="http://www.jstor.org/stable/24524919">Om potensproduktsummer</a> [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. There are errors in the last two rows of his table.

%F a(n)= (n+5)*a(n-1) + (n+1)*A002538(n+1), n>=1, a(0)=1.

%F Recurrence: (n-1)*n^2*a(n) = (n-1)*(3*n^3 + 12*n^2 + 6*n + 1)*a(n-1) - (n+1)*(3*n^4 + 15*n^3 + 13*n^2 - 15*n - 4)*a(n-2) + n*(n+1)^3*(n+2)*(n+3)*a(n-3). - _Vaclav Kotesovec_, May 24 2014

%F a(n) ~ n! * n^5 * log(n) * (log(n)*(1/2+181/(24*n)) + gamma*(1+181/(12*n)) - 2 - 65/(3*n)), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, May 24 2014

%e For instance, a(1) = 22 because among the 7!! = 105 permutations of {1,1,2,2,3,3,4,4} selected according to the definition of Eulerian numbers of the second kind, only 22 contain n = 1 descent, namely : 11223443, 11224433, 11233244, 11233442, 11244233, 11332244, 11334422, 11442233, 12213344, 12233144, 12233441, 12244133, 13312244, 13344122, 14412233, 22113344, 22331144, 22334411, 22441133, 33112244, 33441122, 44112233. - _Jean-François Alcover_, Mar 28 2011

%t b[1]=1; b[2]=22; b[n_] := b[n] = ((n-1)*(n-1)!*n^3 - (n+2)*(n+3)*b[n-2]*n + (n*(2*n+5)-4)*b[n-1]) / (n-1); a[n_] := b[n+1]; Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Mar 23 2011, updated Oct 12 2015 *)

%o (PARI) {a=vector(30,n,1);a[2]=22;for(n=3,#a,a[n]=(n-1)!*n^3+((n*(2*n+5)-4)*a[n-1] - n*(n+2)*(n+3)*a[n-2])/(n-1));a} \\ Uses offet 1 for technical reasons. - _M. F. Hasler_, Sep 19 2015

%Y 3rd diagonal of A008517, third column of A112007.

%K nonn,nice,easy

%O 0,2

%A _N. J. A. Sloane_ , _Simon Plouffe_, _Robert G. Wilson v_, _Mira Bernstein_

%E Formulas adapted for offset 0 by _Vaclav Kotesovec_, May 24 2014

%E More terms from _M. F. Hasler_, Sep 19 2015