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%I #12 Oct 30 2018 20:17:54
%S 12,43,108,228,431,753,1239,1944,2934,4287,6094,8460,11505,15365,
%T 20193,26160,33456,42291,52896,65524,80451,97977,118427,142152,169530,
%U 200967,236898,277788,324133,376461
%N Sixth column (r=5) of FS(3) staircase array A062745.
%C In the Frey-Sellers reference this sequence is called {(n+3) over 5}_{2}, n >= 0.
%H Colin Barker, <a href="/A062749/b062749.txt">Table of n, a(n) for n = 0..1000</a>
%H D. D. Frey and J. A. Sellers, <a href="http://www.fq.math.ca/Scanned/39-2/frey.pdf">Generalizing Bailey's generalization of the Catalan numbers</a>, The Fibonacci Quarterly, 39 (2001) 142-148.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = A062745(n+3, 5)= -3+binomial(n+4, 3)*(n^2+16*n+75)/20 = (n+1)*(n^4+24*n^3+221*n^2+894*n+1440)/5!.
%F G.f.: N(3;2, x)/(1-x)^6 with N(3;2, x)= 12-29*x+30*x^2-15*x^3+3*x^4, polynomial of the third row of A062746.
%F From _Colin Barker_, Oct 30 2018: (Start)
%F G.f.: (12 - 29*x + 30*x^2 - 15*x^3 + 3*x^4) / (1 - x)^6.
%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.
%F (End)
%p seq(coeff(series((3*x^4-15*x^3+30*x^2-29*x+12)/(1-x)^6,x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Oct 30 2018
%o (PARI) Vec((12 - 29*x + 30*x^2 - 15*x^3 + 3*x^4) / (1 - x)^6 + O(x^40)) \\ _Colin Barker_, Oct 30 2018
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Jul 12 2001
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