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A005587 a(n) = n*(n+5)*(n+6)*(n+7)/24.
(Formerly M4929)
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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].

R. K. Guy, Letter to N. J. A. Sloane, Feb 1988

R. 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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

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 *)

PROG

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

(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

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.

Sequence in context: A208359 A245629 A163756 * A244101 A212514 A292051

Adjacent sequences:  A005584 A005585 A005586 * A005588 A005589 A005590

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

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

STATUS

approved

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Last modified November 13 16:02 EST 2018. Contains 317149 sequences. (Running on oeis4.)