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%I M4265 #41 Feb 27 2019 20:56:41
%S 0,6,58,328,1452,5610,19950,67260,218848,695038,2170626,6699696,
%T 20507988,62407890,189123286,571432036,1722945672,5187185766,
%U 15600353130,46882846680,140820504700,422822222266,1269221639358,3809241974028,11431014253872,34299887862990
%N Second-order Eulerian numbers <<n,2>>.
%C See A008517 for the definition of second-order Eulerian numbers.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition. Addison-Wesley, Reading, MA, 1994, p. 270.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A004301/b004301.txt">Table of n, a(n) for n = 2..1000</a>
%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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eulerian_number#Eulerian_numbers_of_the_second_kind">Eulerian numbers of the second kind</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,82,-91,52,-12).
%F From _Michael Somos_, Oct 13 2002: (Start)
%F G.f.: x^3(6-2x-12x^2)/((1-x)^3(1-2x)^2(1-3x)).
%F a(n) = A008517(n, 3) = (9*3^n - (12+8*n)*2^n + (3+6*n+4*n^2))/2. (End)
%F a(n) = Sum_{k=0..n-3} (-1)^(n+k)*binomial(2*n+1, k)*Stirling1(2*n-k-2, n-k-2). - _Johannes W. Meijer_, Oct 16 2009
%e G.f. = 6*x^3 + 58*x^4 + 328*x^5 + 1452*x^6 + 5610*x^7 + 19950*x^8 + ...
%t LinearRecurrence[{10, -40, 82, -91, 52, -12}, {0, 6, 58, 328, 1452, 5610}, 26] (* _Jean-François Alcover_, Feb 27 2019 *)
%o (PARI) {a(n) = if( n<0, 0, (9*3^n - (12 + 8*n)*2^n + (3 + 6*n + 4*n^2))/2)}; /* _Michael Somos_, Oct 13 2002 */
%Y 3rd column of A008517.
%Y 2nd column of A201637.
%Y Equals the fourth right hand column of triangle A163936. - _Johannes W. Meijer_, Oct 16 2009
%K nonn,easy
%O 2,2
%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_
%E Edited by _Olivier Gérard_, Mar 28 2011
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