A101091 Fourth partial sums of fourth powers (A000583).

1, 20, 155, 760, 2814, 8592, 22770, 54120, 117975, 239668, 459173, 837200, 1463020, 2464320, 4019412, 6372144, 9849885, 14884980, 22040095, 32037896, 45795530, 64464400, 89475750, 122592600, 165968595, 222214356, 294471945, 386498080

Offset: 1

Author

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004

Keywords

• easy
• nonn

Links

• Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Dead link]
• Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A000583, A101090.

Programs

Mathematica

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 *)

Formula

a(n) = n*(1 + n)*(2 + n)^2*(3 + n)*(4 + n)*(-1 + 3*n*(4 + n))/5040.

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

G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^9. - Colin Barker, Apr 04 2012

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

Extensions

Edited by Ralf Stephan, Dec 16 2004