A101099 Third partial sums of fifth powers (A000584).

1, 35, 345, 1955, 7990, 26226, 73470, 182490, 412335, 863005, 1695551, 3158805, 5624060, 9629140, 15933420, 25585476, 40005165, 61082055, 91292245, 133835735, 192796626, 273328550, 381867850, 526377150, 716622075, 964484001, 1284311835

Offset: 1

Author

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• 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. A000584.

Programs

Magma

[n*(1+n)*(2+n)*(3+n)*(-1+n*(2+n))*(2+n*(4+n))/336: n in [1..30]]; // G. C. Greubel, Dec 01 2018

Mathematica

Nest[Accumulate[#]&,Range[30]^5,3] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,35,345,1955,7990,26226,73470,182490,412335},30] (* Harvey P. Dale, Feb 20 2015 *)

PARI

vector(30, n, n*(1+n)*(2+n)*(3+n)*(-1+n*(2+n))*(2+n*(4+n))/336) \\ G. C. Greubel, Dec 01 2018

Sage

[n*(1+n)*(2+n)*(3+n)*(-1+n*(2+n))*(2+n*(4+n))/336 for n in (1..30)] # G. C. Greubel, Dec 01 2018

Formula

a(n) = n*(1 + n)*(2 + n)*(3 + n)*(-1 + n*(2 + n))*(2 + n*(4 + n))/336.

G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^9. - Colin Barker, Apr 16 2012

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, Feb 20 2015

E.g.f.: x*(336 + 5544*x + 13608*x^2 + 10934*x^3 + 3696*x^4 + 574*x^5 + 40*x^6 + x^7)*exp(x)/336. - G. C. Greubel, Dec 01 2018

Sum_{n>=1} 1/a(n) = 224/3 - 60*sqrt(2)*Pi*cot(sqrt(2)*Pi). - Amiram Eldar, Jan 27 2022

Extensions

Edited by Ralf Stephan, Dec 16 2004