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Sum of the first k-1 numbers in the k-th column of the natural number array A000027, by antidiagonals.
3

%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