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%I #17 Jan 26 2022 02:30:13
%S 1,20,155,760,2814,8592,22770,54120,117975,239668,459173,837200,
%T 1463020,2464320,4019412,6372144,9849885,14884980,22040095,32037896,
%U 45795530,64464400,89475750,122592600,165968595,222214356,294471945,386498080
%N Fourth partial sums of fourth powers (A000583).
%H Cecilia Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Dead link]
%H Cecilia Rossiter, <a href="/A101096/a101096.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Cached copy, May 15 2013]
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = n*(1 + n)*(2 + n)^2*(3 + n)*(4 + n)*(-1 + 3*n*(4 + n))/5040.
%F a(1)=1, a(2)=20, a(3)=155, a(4)=760, a(5)=2814, a(6)=8592, a(7)=22770, a(8)=54120, a(9)=117975, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)- 126*a(n-4)+ 126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - _Harvey P. Dale_, Dec 30 2011
%F G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^9. - _Colin Barker_, Apr 04 2012
%F Sum_{n>=1} 1/a(n) = 3934693/3380 - 210*Pi^2/13 - (2268/13)*sqrt(3/13)*Pi*cot(sqrt(13/3)*Pi). - _Amiram Eldar_, Jan 26 2022
%t Nest[Accumulate,Range[30]^4,4] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{1,20,155,760,2814,8592,22770,54120,117975},30] (* _Harvey P. Dale_, Dec 30 2011 *)
%Y Cf. A000583, A101090.
%K easy,nonn
%O 1,2
%A Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004
%E Edited by _Ralf Stephan_, Dec 16 2004
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