cp's OEIS Frontend

A185788 Sum of the first k-1 numbers in the k-th column of the natural number array A000027, by antidiagonals.

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