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

1,2

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).

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

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A000584.

Sequence in context: A372151 A027792 A163935 * A027803 A267749 A073567

Adjacent sequences: A101096 A101097 A101098 * A101100 A101101 A101102

Keyword

nonn,easy

Author

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

Extensions

Edited by Ralf Stephan, Dec 16 2004

Status

approved