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A049401
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Number of Young tableaux of height <= 5.
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15
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1, 1, 2, 4, 10, 26, 75, 225, 715, 2347, 7990, 27908, 99991, 365587, 1362310, 5159208, 19831101, 77233517, 304423574, 1212962072, 4881181036, 19821471956, 81165639197, 334925706659, 1391935877463, 5823186349671, 24511802558326, 103772782048252, 441696903185704
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OFFSET
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0,3
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COMMENTS
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Also the number of n-length words w over alphabet {a,b,c,d,e} such that for every prefix z of w we have #(z,a) >= #(z,b) >= #(z,c) >= #(z,d) >= #(z,e), where #(z,x) counts the letters x in word z. The a(5) = 26 words are: aaaaa, aaaab, aaaba, aabaa, abaaa, aaabb, aabab, abaab, aabba, ababa, aaabc, aabac, abaac, aabca, abaca, abcaa, aabbc, ababc, aabcb, abacb, abcab, aabcd, abacd, abcad, abcda, abcde. - Alois P. Heinz, May 30 2012
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(b), y_5(n), p. 452.
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LINKS
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FORMULA
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E.g.f.: e^x*(BesselI(0, 2*x)^2 - BesselI(0, 2*x)*BesselI(2, 2*x) - BesselI(0, 2*x)*BesselI(4, 2*x) - BesselI(1, 2*x)^2 + 2*BesselI(1, 2*x)*BesselI(3, 2*x) + BesselI(2, 2*x)*BesselI(4, 2*x) - BesselI(3, 2*x)^2) (BesselI = modified Bessel function of first kind).
D-finite with recurrence (n+6)*(n+4)*a(n) +(-3*n^2-17*n-15)*a(n-1) -(13*n+9)*(n-1)*a(n-2) +15*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 23 2021
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MAPLE
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a:= proc(n) option remember;
`if`(n<3, [1, 1, 2][n+1], ((3*n^2+17*n+15)*a(n-1)
+(n-1)*(13*n+9)*a(n-2) -15*(n-1)*(n-2)*a(n-3)) /
((n+4)*(n+6)))
end:
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MATHEMATICA
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a[n_] := a[n] = If[n<3, {1, 1, 2}[[n+1]], ((3*n^2+17*n+15)*a[n-1] + (n-1)*(13*n+9)*a[n-2] - 15*(n-1)*(n-2)*a[n-3]) / ((n+4)*(n+6))]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
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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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More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
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STATUS
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approved
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