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A002539 Eulerian numbers of the second kind: <<n+3, n>>.
(Formerly M5126 N2221)
9
1, 22, 328, 4400, 58140, 785304, 11026296, 162186912, 2507481216, 40788301824, 697929436800, 12550904017920, 236908271543040, 4687098165573120, 97049168010017280, 2099830209402931200, 47405948832458496000, 1115089078488795648000, 27290469545695931904000, 694002594415741341696000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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.
The sequence gives the Eulerian numbers <<3,0>>, <<4,1>>, <<5,2>>, <<6,3>>, ... (and in particular the offset is 0).
REFERENCES
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2nd edition; Addison-Wesley, 1994, pp. 270-271.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
L. Carlitz, Some numbers related to the Stirling numbers of the first and second kind, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., Numbers 544-576 (1976): 49-55. [Annotated scanned copy. The triangle is A008517.]
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]
O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. There are errors in the last two rows of his table.
FORMULA
a(n)= (n+5)*a(n-1) + (n+1)*A002538(n+1), n>=1, a(0)=1.
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
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
EXAMPLE
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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
3rd diagonal of A008517, third column of A112007.
Sequence in context: A025941 A125479 A047868 * A230965 A019958 A021644
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Formulas adapted for offset 0 by Vaclav Kotesovec, May 24 2014
More terms from M. F. Hasler, Sep 19 2015
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)