LINKS
<a href="/index/Rec">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
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Revision History for A005720
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LINKS
<a href="/index/Rea#recLCCRec">Index to sequences with linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
STATUS
proposed
approved
STATUS
editing
proposed
FORMULA
G.f.: (x^2)*(1+3*x-5*x^2+2*x^3)/(1-x)^7 (numerator polynomial is N4(6, x) from A063421.).
a(0)=1, a(1)=10, a(2)=44, a(3)=135, a(4)=336, a(5)=728, a(6)=1428, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7) [From . - _Harvey P. Dale, June _, Jun 23 2011]
a(n) = binomial(n,2) + 7*binomial(n,3) + 10*binomial(n,4) + 5*binomial(n,5) + binomial(n,6) (see comment in A071675). - _Vladimir Shevelev _ and _Peter J. C. Moses, _, Jun 22 2012
MATHEMATICA
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 10, 44, 135, 336, 728, 1428}, 40] (* or *) Table[Binomial[n+1, 3] (n^3+15n^2+86n-120)/120, {n, 2, 41}] (* Harvey P. Dale, June Jun 23 2011 *)
AUTHOR
STATUS
approved
editing
LINKS
Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices séries génératrices et Quelques Conjecturesquelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.
MATHEMATICA
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 10, 44, 135, 336, 728, 1428}, 40] (* or *) Table[Binomial[n+1, 3] (n^3+15n^2+86n-120)/120, {n, 2, 41}] (* From _Harvey P. Dale, _, June 23 2011 *)
LINKS
Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} Université du Qu\'{e}bec \`{a} Montr\'{e}al, Québec à Montréal, 1992.
Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} Université du Qu\'{e}bec \`{a} Montr\'{e}al, Québec à Montréal, 1992.
LINKS
_Simon Plouffe_, , <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
LINKS
_Simon Plouffe_, , <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.