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A004301
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Second-order Eulerian numbers <<n,2>>.
(Formerly M4265)
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7
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0, 6, 58, 328, 1452, 5610, 19950, 67260, 218848, 695038, 2170626, 6699696, 20507988, 62407890, 189123286, 571432036, 1722945672, 5187185766, 15600353130, 46882846680, 140820504700, 422822222266, 1269221639358, 3809241974028, 11431014253872, 34299887862990
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OFFSET
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2,2
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COMMENTS
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See A008517 for the definition of second-order Eulerian numbers.
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition. Addison-Wesley, Reading, MA, 1994, p. 270.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: x^3(6-2x-12x^2)/((1-x)^3(1-2x)^2(1-3x)).
a(n) = A008517(n, 3) = (9*3^n - (12+8*n)*2^n + (3+6*n+4*n^2))/2. (End)
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
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EXAMPLE
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G.f. = 6*x^3 + 58*x^4 + 328*x^5 + 1452*x^6 + 5610*x^7 + 19950*x^8 + ...
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MATHEMATICA
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LinearRecurrence[{10, -40, 82, -91, 52, -12}, {0, 6, 58, 328, 1452, 5610}, 26] (* Jean-François Alcover, Feb 27 2019 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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