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A335648 Partial sums of A006010.

%I A335648 #36 Sep 08 2022 08:46:25
%S A335648 0,1,6,26,78,195,420,820,1476,2501,4026,6222,9282,13447,18984,26216,
%T A335648 35496,47241,61902,80002,102102,128843,160908,199068,244140,297037,
%U A335648 358722,430262,512778,607503,715728,838864,978384,1135889,1313046,1511658,1733598,1980883,2255604
%N A335648 Partial sums of A006010.
%H A335648 Stefano Spezia, <a href="/A335648/b335648.txt">Table of n, a(n) for n = 0..10000</a>
%H A335648 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-4,10,-4,-4,4,-1).
%H A335648 <a href="/index/Par#partial">Index entries for partial sums</a>.
%F A335648 a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80.
%F A335648 O.g.f.: x*(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^6*(1 + x)^2).
%F A335648 E.g.f.: (cosh(x) - sinh(x))*(-5 + 5*x + (5 + 65*x + 180*x^2 + 130*x^3 + 30*x^4 + 2*x^5)*(cosh(2*x) + sinh(2*x)))/80.
%F A335648 a(n) = 4*a(n-1) - 4*a(n-2) - 4*a(n-3) + 10*a(n-4) - 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - a(n-8) for n > 7.
%F A335648 a(2*n-1) = n*A053755(n)/5 for n > 0.
%F A335648 a(2*n) = n*A005408(n)*A059722(n-1)/5.
%F A335648 a(2*n+1) - a(2*n-1) = A001844(n)^2 = A007204(n) for n > 0.
%F A335648 a(2*n) - a(2*n-2) = 2*A000290(n)*A058331(n) for n > 0.
%t A335648 Table[(1+n)(5-5(-1)^n+8n+12n^2+8n^3+2n^4)/80,{n,0,38}]
%o A335648 (Magma) I:=[0, 1, 6, 26, 78, 195, 420, 820]; [n le 8 select I[n] else 4*Self(n-1)-4*Self(n-2)-4*Self(n-3)+10*Self(n-4)-4*Self(n-5)-4*Self(n-6)+4*Self(n-7)-Self(n-8): n in [1..39]];
%o A335648 (PARI) a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80;
%o A335648 (Sage) (x*(1+2*x+6*x^2+2*x^3+x^4)/((1-x)^6*(1+x)^2)).series(x, 39).coefficients(x, False)
%Y A335648 Cf. A000290, A001844, A005408, A007204, A053755, A058331, A059722.
%Y A335648 Cf. A006010 (1st differences), A186424 (3rd differences), A317614 (2nd differences).
%K A335648 nonn,easy
%O A335648 0,3
%A A335648 _Stefano Spezia_, Jun 15 2020