Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #36 Mar 24 2019 07:50:57
%S 0,0,1,3,6,10,15,21,29,45,90,220,561,1365,3095,6555,13110,25126,46971,
%T 87381,164921,320001,640002,1309528,2707629,5592405,11450531,23166783,
%U 46333566,91869970,181348455,357913941,708653429,1410132405,2820264810,5662052980
%N Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).
%C Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences:
%C 0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n): after 0, it is A192080.
%C 0, 0, 0, 0, 1, 5, 15, 35, 70, 126,... e(n)
%C 0, 0, 0, 1, 4, 10, 20, 35, 56, 85,... f(n)
%C 0, 0, 1, 3, 6, 10, 15, 21, 29, 45,... a(n)
%C 0, 1, 2, 3, 4, 5, 6, 8, 16, 45,... b(n)
%C 1, 1, 1, 1, 1, 1, 2, 8, 29, 85,... c(n)
%C 0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n).
%C a(n) + d(n) = A024495(n),
%C b(n) + e(n) = A131708(n),
%C c(n) + f(n) = A024493(n).
%C a(n) - d(n) = 0, 0, 1, 3, 6, 9, 9, 0,... A057083(n-2)
%C b(n) - e(n) = 0, 1, 2, 3, 3, 0, -9, -27,... A057682(n)
%C c(n) - f(n) = 1, 1, 1, 0, -3, -9, -18, -27,... A057681(n)
%C d(n) - a(n) = 0, 0, -1, -3, -6, -9, -9, 0,... -A057083(n-2)
%C e(n) - b(n) = 0, -1, -2, -3, -3, 0, 9, 27,... -A057682(n)
%C f(n) - c(n) = -1, -1, -1, 0, 3, 9, 18, 27,... -A057681(n).
%C The first column is A131531(n).
%C The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9 A029898(n)?
%H Seiichi Manyama, <a href="/A227430/b227430.txt">Table of n, a(n) for n = 0..3000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6).
%F a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10.
%F a(n) = A024495(n) - A192080(n-5) for n>4.
%F G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - _Ralf Stephan_, Jul 13 2013
%F a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - _Seiichi Manyama_, Mar 23 2019
%e a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.
%t Join[{0},LinearRecurrence[{6,-15,20,-15,6},{0,1,3,6,10},40]] (* _Harvey P. Dale_, Dec 17 2014 *)
%o (PARI) {a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ _Seiichi Manyama_, Mar 23 2019
%K nonn,easy
%O 0,4
%A _Paul Curtz_, Jul 11 2013
%E Definition uses the g.f. of Ralf Stephan.
%E More terms from _Harvey P. Dale_, Dec 17 2014