A101854 a(n) = n*(n+1)*(n^2 + 21*n + 50)/24.

6, 24, 61, 125, 225, 371, 574, 846, 1200, 1650, 2211, 2899, 3731, 4725, 5900, 7276, 8874, 10716, 12825, 15225, 17941, 20999, 24426, 28250, 32500, 37206, 42399, 48111, 54375, 61225, 68696, 76824, 85646, 95200, 105525, 116661, 128649, 141531, 155350

Offset: 1

Author

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 18 2004

Keywords

• easy
• nonn

Comments

5th partial summation within series as series accumulate n times from an initial sequence of Euler Triangle's row 3: 1,4,1.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]
• C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

5th row of the array shown in A101853.

Partial sums of A101853.

Programs

Mathematica

Table[25 n/12+(71n^2)/24+(11n^3)/12+n^4/24,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{6,24,61,125,225},40] (* Harvey P. Dale, Nov 05 2011 *)

Formula

G.f.: x*(6 - 6*x + x^2)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; checked and corrected by R. J. Mathar, Sep 16 2009

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 5. - Harvey P. Dale, Nov 05 2011

E.g.f.: exp(x)*x*(144 + 144*x + 28*x^2 + x^3)/24. - Stefano Spezia, Oct 14 2022

Extensions

Formula moved to be the definition by Eric M. Schmidt, Dec 12 2013