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A005587
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a(n) = n*(n+5)*(n+6)*(n+7)/24.
(Formerly M4929)
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10
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0, 14, 42, 90, 165, 275, 429, 637, 910, 1260, 1700, 2244, 2907, 3705, 4655, 5775, 7084, 8602, 10350, 12350, 14625, 17199, 20097, 23345, 26970, 31000, 35464, 40392, 45815, 51765, 58275, 65379, 73112, 81510, 90610, 100450, 111069, 122507, 134805, 148005
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OFFSET
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0,2
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COMMENTS
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a(n) = number of Standard Young Tableaux of shape (n+3,4). - David Callan, Aug 17 2004
a(n) for n > 0 is the number of n-extended coalescent histories for a matching caterpillar gene tree and species tree with 5 leaves. - Noah A Rosenberg, Jun 16 2022
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
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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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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G.f.: (14 - 28*x + 20*x^2 - 5*x^3) / (1 - x)^5.
E.g.f.: (1/24)*x*(336 + 168*x + 24*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 01 2017
Sum_{n>=1} 1/a(n) = 153/1225.
Sum_{n>=1} (-1)^(n+1)/a(n) = 288*log(2)/35 - 20759/3675. (End)
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MAPLE
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seq(numbperm(n, 4)/24-numbperm(n, 3)/6, n=7..46); # Zerinvary Lajos, May 20 2008
a:=n->(sum(numbcomp(n, 4), j=9..n)):seq(a(n)/4, n=8..47); # Zerinvary Lajos, Aug 26 2008
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MATHEMATICA
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LinearRecurrence[{5, -10, 10, -5, 1}, {0, 14, 42, 90, 165}, 40] (* Harvey P. Dale, Aug 17 2017 *)
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PROG
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(PARI) x='x+O('x^50); concat([0], Vec((14 - 28*x + 20*x^2 - 5*x^3) / (1 - x)^5)) \\ G. C. Greubel, Jul 01 2017
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CROSSREFS
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Fifth diagonal of Catalan triangle A033184. Fifth column of Catalan triangle A009766.
Numerator polynomial 14 - 28x + 20x^2 - 5x^3 from fourth row of triangle A062991.
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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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M4929 (this sequence) and M4930 were the same.
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STATUS
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approved
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