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A112851 a(n) = (n-1)*n*(n+1)*(n+2)*(2*n+1)/40.

%I A112851 #52 Feb 16 2025 08:32:59
%S A112851 0,0,3,21,81,231,546,1134,2142,3762,6237,9867,15015,22113,31668,44268,
%T A112851 60588,81396,107559,140049,179949,228459,286902,356730,439530,537030,
%U A112851 651105,783783,937251,1113861,1316136,1546776,1808664,2104872,2438667,2813517,3233097
%N A112851 a(n) = (n-1)*n*(n+1)*(n+2)*(2*n+1)/40.
%C A112851 A112851 is the fourth sequence in A112852.
%C A112851 Also the Wiener index of the (n-1)-triangular grid graph (indexed so the 0-triangular grid graph is the singleton). - _Eric W. Weisstein_, Sep 08 2017
%D A112851 L. B. W. Jolley, Summation of Series, Dover. N.Y., 1961, eq. (54), page 10.
%H A112851 Vincenzo Librandi, <a href="/A112851/b112851.txt">Table of n, a(n) for n = 0..1000</a>
%H A112851 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGridGraph.html">Triangular Grid Graph</a>.
%H A112851 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%H A112851 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A112851 4*a(n+1) = 1*2^2*3 + 2*3^2*4 + 3*4^2*5 + ... (n terms). [Jolley]
%F A112851 a(n) = Sum_{i=0..n} A000217(i-1)*A000217(i), where A000217(-1)=0. - _Bruno Berselli_, Feb 05 2014
%F A112851 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. - _Harvey P. Dale_, Oct 28 2014
%F A112851 G.f.: 3*x^2*(1 + x) / (1 - x)^6. - _Colin Barker_, Sep 08 2017
%F A112851 a(n) = (1/2) * Sum_{k=0..n} C(k^2,2). - _Wesley Ivan Hurt_, Sep 23 2017
%F A112851 a(n) = Sum_{i=0..n} A000217(i)*A033428(n-i). - _Bruno Berselli_, Mar 06 2018
%F A112851 From _Amiram Eldar_, Feb 15 2022: (Start)
%F A112851 Sum_{n>=2} 1/a(n) = 40*(16*log(2) - 11)/9.
%F A112851 Sum_{n>=2} (-1)^n/a(n) = 20*(8*Pi - 25)/9. (End)
%p A112851 a:=n->sum(j^4-j^2, j=0..n)/4: seq(a(n), n=0..36); # _Zerinvary Lajos_, May 08 2008
%t A112851 Table[(n - 1) n (n + 1)(n + 2)(2 n + 1)/40, {n, 0, 30}] (* Josh Locker *)
%t A112851 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 3, 21, 81, 231}, 40] (* _Harvey P. Dale_, Oct 28 2014 *)
%o A112851 (Magma) [(n-1)*n*(n+1)*(n+2)*(2*n+1)/40: n in [0..40]]; // _Vincenzo Librandi_, Feb 06 2014
%o A112851 (PARI) for(n=0,50, print1((n-1)*n*(n+1)*(n+2)*(2*n+1)/40, ", ")) \\ _G. C. Greubel_, Jul 23 2017
%o A112851 (PARI) concat(vector(2), Vec(3*x^2*(1 + x) / (1 - x)^6 + O(x^30))) \\ _Colin Barker_, Sep 08 2017
%Y A112851 Partial sums of sequence A006011.
%Y A112851 Cf. A000217, A033428, A112851, A112852.
%K A112851 nonn,easy
%O A112851 0,3
%A A112851 _Alford Arnold_, Sep 24 2005
%E A112851 More terms from Josh Locker (jlocker(AT)mail.rochester.edu) and Michael W. Motily (mwm5036(AT)psu.edu), Oct 04 2005