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%I #25 Dec 11 2015 17:41:32
%S 1,5,10,22,34,58,80,120,155,215,266,350,420,532,624,768,885,1065,1210,
%T 1430,1606,1870,2080,2392,2639,3003,3290,3710,4040,4520,4896,5440,
%U 5865,6477,6954,7638,8170,8930,9520,10360,11011,11935,12650,13662,14444
%N Add column entries of the table with rows (1,2,0,0...), (0,3,4,5,0,0...), (0,0,6,7,8,9,0,0...), (0,0,0,10,11,12,13,14,0,0...), ...
%H Harry J. Smith, <a href="/A064694/b064694.txt">Table of n, a(n) for n=1..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F a(2n-1) = n*(n+1)*(7*n-4)/6 (see A007584), a(2n) = n*(n+1)*(7*n+8)/6.
%F a(n) = sum{k=0..floor((n+1)/2), (n-k+1)ceiling((n-k+1)/2)+k+if(mod(n-k+1, 2) =0, ceiling((n-k+1)/2), 0)}. - _Paul Barry_, Aug 25 2004
%F a(n) = sum{k=0..floor(n/2), C(n-k+1,2)+k}; - _Paul Barry_, Jul 23 2008
%F a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(14*n+15+17*(-1)^n)/384. - _Luce ETIENNE_, Feb 17 2015
%F From _Colin Barker_, Feb 17 2015: (Start)
%F a(n) = (7*n^3+30*n^2+32*n)/48 for n even.
%F a(n) = (7*n^3+27*n^2+17*n-3)/48 for n odd.
%F G.f.: x*(2*x^2+4*x+1) / ((x-1)^4*(x+1)^3).
%F (End)
%e a(1)=1, a(2)=2+3=5, a(3)=4+6=10, a(4)=5+7+10=22.
%t Table[ Sum[ Binomial[n-k+1, 2] + k, {k, 0, Floor[n/2]}], {n, 1, 45}] (* _Jean-François Alcover_, Sep 16 2013 *)
%t LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,5,10,22,34,58,80},50] (* _Harvey P. Dale_, Dec 11 2015 *)
%o (PARI) { for (n=1, 1000, a=sum(k=0, n\2, binomial(n - k + 1, 2) + k); write("b064694.txt", n, " ", a) ) } \\ _Harry J. Smith_, Sep 22 2009
%o (PARI) Vec(x*(2*x^2+4*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ _Colin Barker_, Feb 17 2015
%Y Cf. A007584.
%K nonn,easy,nice
%O 1,2
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 12 2001
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