A005587 a(n) = n*(n+5)*(n+6)*(n+7)/24.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) = number of Standard Young Tableaux of shape (n+3,4). - David Callan, Aug 17 2004

a(n) = A214292(n+6,3). - Reinhard Zumkeller, Jul 12 2012

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

References

• 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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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].
• Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988.
• Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• N. A. Rosenberg, Counting coalescent histories, J. Comput. Biol. 14 (2007), 360-377.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

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.

Programs

Magma

[n*(n+5)*(n+6)*(n+7)/24: n in [0..40]]; // Vincenzo Librandi, Mar 20 2013

Maple

A005587:=z*(-14+28*z-20*z**2+5*z**3)/(z-1)**5; # Simon Plouffe in his 1992 dissertation
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

Mathematica

Table[n (n + 5) (n + 6) (n + 7)/24, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{0,14,42,90,165},40] (* Harvey P. Dale, Aug 17 2017 *)

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

Formula

G.f.: (14 - 28*x + 20*x^2 - 5*x^3) / (1 - x)^5.

a(n) = C(7+n, 4) - C(7+n, 3). - Zerinvary Lajos, Dec 09 2005

E.g.f.: (1/24)*x*(336 + 168*x + 24*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 01 2017

From Amiram Eldar, Jun 28 2022: (Start)

Sum_{n>=1} 1/a(n) = 153/1225.

Sum_{n>=1} (-1)^(n+1)/a(n) = 288*log(2)/35 - 20759/3675. (End)

a(n) = A024191(n+1)-5. - R. J. Mathar, Nov 22 2024

Extensions

M4929 (this sequence) and M4930 were the same.

More terms from Matthew Conroy, Jan 16 2006

Plouffe Maple line edited by N. J. A. Sloane, May 13 2008