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A203552 a(n) = n*(5*n^2 - 3*n + 4) / 6.

%I A203552 #39 Nov 13 2024 16:38:55
%S A203552 0,1,6,20,48,95,166,266,400,573,790,1056,1376,1755,2198,2710,3296,
%T A203552 3961,4710,5548,6480,7511,8646,9890,11248,12725,14326,16056,17920,
%U A203552 19923,22070,24366,26816,29425,32198,35140,38256,41551,45030,48698
%N A203552 a(n) = n*(5*n^2 - 3*n + 4) / 6.
%H A203552 Vincenzo Librandi, <a href="/A203552/b203552.txt">Table of n, a(n) for n = 0..10000</a>
%H A203552 Philip Boalch, <a href="https://arxiv.org/abs/2410.23358">Counting the fission trees and nonabelian Hodge graphs</a>, arXiv:2410.23358 [math.AG], 2024. See p. 11.
%H A203552 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A203552 a(n) = Sum_{k = 1..n} A(k-1, n-k) where A(i, j) = i^2 + i*j + j^2 + i + j + 1.
%F A203552 G.f.: x * (1 + 2*x + 2*x^2) / (1 - x)^4.
%F A203552 a( n) = -A203551(-n) for all n in Z.
%F A203552 a(n)-a(n-1) = A134238(n). - _Bruno Berselli_, Jan 03 2012
%F A203552 a(n) = 4*A000125(n) + 2*A000125(n+1) - A000125(n+3). - _Ivan N. Ianakiev_, Aug 21 2013
%F A203552 E.g.f.: x*(5*x^2 + 12*x + 6)*exp(x)/6. - _G. C. Greubel_, Aug 12 2018
%e A203552 G.f. = x + 6*x^2 + 20*x^3 + 48*x^4 + 95*x^5 + 166*x^6 + 266*x^7 + 400*x^8 + ...
%t A203552 LinearRecurrence[{4,-6,4,-1},{0,1,6,20},40] (* _Vincenzo Librandi_, Jan 07 2012 *)
%o A203552 (PARI) {a(n) = n * (5*n^2 - 3*n + 4) / 6};
%o A203552 (Magma) I:=[0, 1, 6, 20]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jan 07 2012
%Y A203552 Cf. A000125, A134238, A203551.
%K A203552 nonn,easy
%O A203552 0,3
%A A203552 _Michael Somos_, Jan 02 2012