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a(n) = Sum_{i=1..n} binomial(i+1,2)^3.
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%I #35 Sep 08 2022 08:45:11

%S 1,28,244,1244,4619,13880,35832,82488,173613,339988,627484,1102036,

%T 1855607,3013232,4741232,7256688,10838265,15838476,22697476,31958476,

%U 44284867,60479144,81503720,108503720,142831845,186075396,240085548,307008964,389321839

%N a(n) = Sum_{i=1..n} binomial(i+1,2)^3.

%D Elisabeth Busser and Gilles Cohen, Neuro-Logies - "Chercher, jouer, trouver", La Recherche, April 1999, No. 319, page 97.

%H G. C. Greubel, <a href="/A085438/b085438.txt">Table of n, a(n) for n = 1..5000</a>

%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) = (90*n^7 +630*n^6 +1638*n^5 +1890*n^4+ 840*n^3 -48*n)/7!.

%F a(n) = (C(n+2, 3)/35)*(35 +210*C(n-1, 1) +399*C(n-1, 2) +315*C(n-1, 3) +90*C(n-1, 4)).

%F G.f.: x*(x^4+20*x^3+48*x^2+20*x+1) / (x-1)^8. - _Colin Barker_, May 02 2014

%e a(10) = (90*(10^7)+630*(10^6)+1638*(10^5)+1890*(10^4)+840*(10^3)-48*(10))/5040 = 339988.

%t Table[(90*n^7 + 630*n^6 + 1638*n^5 + 1890*n^4 + 840*n^3 - 48*n)/7!, {n, 1, 50}] (* _G. C. Greubel_, Nov 22 2017 *)

%o (PARI) Vec(x*(x^4+20*x^3+48*x^2+20*x+1)/(x-1)^8 + O(x^100)) \\ _Colin Barker_, May 02 2014

%o (PARI) a(n) = sum(i=1, n, binomial(i+1, 2)^3); \\ _Michel Marcus_, Nov 22 2017

%o (Magma) [(90*n^7 +630*n^6 +1638*n^5 +1890*n^4+ 840*n^3 -48*n)/ Factorial(7): n in [1..30]]; // _G. C. Greubel_, Nov 22 2017

%Y Column k=3 of A334781.

%Y Cf. A000292, A087127, A024166, A024166, A085439, A085440, A085441, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.

%K easy,nonn

%O 1,2

%A _André F. Labossière_, Jun 30 2003

%E More terms from _Colin Barker_, May 02 2014

%E Formula and example edited by _Colin Barker_, May 02 2014