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%I #45 Feb 16 2022 23:23:46
%S 1,17,117,517,1742,4878,11934,26334,53559,101959,183755,316251,523276,
%T 836876,1299276,1965132,2904093,4203693,5972593,8344193,11480634,
%U 15577210,20867210,27627210,36182835,46915011,60266727,76750327
%N a(n) = Sum_(i=1..n) binomial(i+2,3)^2 [ Sequential sums of the tetragonal numbers or "tetras" (pyramidal, square) raised to power 2 (drawn from the 4th diagonal - left or right - of Pascal's Triangle) ].
%C Kekulé numbers for certain benzenoids (see the Cyvin-Gutman reference, p. 243; expression in (13.26) yields same sequence with offset 0). - _Emeric Deutsch_, Aug 02 2005
%C Partial sums of A001249. - _R. J. Mathar_, Aug 19 2008
%H T. D. Noe, <a href="/A086020/b086020.txt">Table of n, a(n) for n = 1..1000</a>
%H S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.
%H John Engbers and Christopher Stocker, <a href="http://epublications.marquette.edu/mscs_fac/456/">Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients</a>, Integers 16 (2016), #A58.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F a(n) = Sum_(i=1..n) binomial(i+2, 3)^2.
%F a(n) = ( C(n+3, 4)/35 )*( 35 + 84*C(n-1, 1) + 70*C(n-1, 2) + 20*C(n-1, 3) ).
%F a(n) = n*(n+1)*(n+2)*(n+3)*(2*n+3)(5*n^2 + 15*n + 1)/2520. - _Emeric Deutsch_, Aug 02 2005
%F O.g.f: x*(1+x)*(1 + 8*x + x^2)/(1-x)^8. - _R. J. Mathar_, Aug 19 2008
%e a(8) = Sum_{i=1..8} binomial(i+2,3)^2 = (20*(8^7) + 210*(8^6) + 854*(8^5) + 1680*(8^4) + 1610*(8^3) + 630*(8^2) + 36*8)/7! = 26334.
%p a:=n->n*(n+1)*(n+2)*(n+3)*(2*n+3)*(5*n^2+15*n+1)/2520: seq(a(n),n=1..31); # _Emeric Deutsch_
%t Accumulate[Binomial[Range[30]+2,3]^2] (* _Harvey P. Dale_, Mar 24 2011 *)
%t LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,17,117,517,1742,4878, 11934, 26334},30] (* _Harvey P. Dale_, Aug 17 2014 *)
%o (PARI) a(n)=n*(n+1)*(n+2)*(n+3)*(2*n+3)*(5*n^2+15*n+1)/2520 \\ _Charles R Greathouse IV_, May 18 2015
%o (Magma) [n*(n+1)*(n+2)*(n+3)*(2*n+3)*(5*n^2+15*n+1)/2520: n in [1..30]]; // _G. C. Greubel_, Nov 22 2017
%Y Cf. A000292, A087127, A024166, A085438, A085439, A085440, A085441, A085442, A000332, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.
%K easy,nice,nonn
%O 1,2
%A _André F. Labossière_, Jul 17 2003
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