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%I #24 Jul 13 2017 03:12:53
%S 0,2,12,37,84,160,272,427,632,894,1220,1617,2092,2652,3304,4055,4912,
%T 5882,6972,8189,9540,11032,12672,14467,16424,18550,20852,23337,26012,
%U 28884,31960,35247,38752,42482,46444,50645,55092,59792,64752,69979,75480,81262,87332,93697,100364,107340,114632,122247,130192,138474
%N Sum of the first k-1 numbers in the k-th column of the natural number array A000027, by antidiagonals.
%C See A185787.
%H G. C. Greubel, <a href="/A185788/b185788.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1)
%F a(n) = (n-1)*(7*n^2 - 11*n + 6)/6. - Corrected by _Manfred Arens_, Mar 11 2016
%F G.f.: x^2*(2+4*x+x^2) / (x-1)^4 . - _R. J. Mathar_, Aug 23 2012
%e Start from
%e 1.....2....4.....7...11...16...22...29...
%e 3.....5....8....12...17...23...30...38...
%e 6.....9...13....18...24...31...39...48...
%e 10...14...19....25...32...40...49...59...
%e 15...20...26....33...41...50...60...71...
%e 21...27...34....42...51...61...72...84...
%e 28...35...43....52...62...73...85...98...
%e Block out all terms starting at and below the main diagonal then sum up the remaining terms.
%e .....2.....4.....7...11...16...22...29...
%e ...........8....12...17...23...30...38...
%e ................18...24...31...39...48...
%e .....................32...40...49...59...
%e ..........................50...60...71...
%e ...............................72...84...
%e ....................................98...
%t f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
%t s[k_]:=Sum[f[n,k],{n,1,k-1}];
%t Factor[s[k]]
%t Table[s[k],{k,1,70}]
%t Table[(n - 1)*(7*n^2 - 11*n + 6)/6, {n, 1, 50}] (* _G. C. Greubel_, Jul 12 2017 *)
%o (PARI) for(n=1,50, print1((n-1)*(7*n^2 - 11*n + 6)/6, ", ")) \\ _G. C. Greubel_, Jul 12 2017
%Y Cf. A000027, A185787, A079824.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Feb 03 2011
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