A101854 - OEIS

%I #29 Oct 15 2022 07:35:10

%S 6,24,61,125,225,371,574,846,1200,1650,2211,2899,3731,4725,5900,7276,

%T 8874,10716,12825,15225,17941,20999,24426,28250,32500,37206,42399,

%U 48111,54375,61225,68696,76824,85646,95200,105525,116661,128649,141531,155350

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

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

%H Harvey P. Dale, <a href="/A101854/b101854.txt">Table of n, a(n) for n = 1..1000</a>

%H C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Dead link]

%H C. Rossiter, <a href="/A101094/a101094_1.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Cached copy, May 15 2013]

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F 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

%F 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

%F E.g.f.: exp(x)*x*(144 + 144*x + 28*x^2 + x^3)/24. - _Stefano Spezia_, Oct 14 2022

%t 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 *)

%Y 5th row of the array shown in A101853.

%Y Partial sums of A101853.

%K easy,nonn

%O 1,1

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

%E Formula moved to be the definition by _Eric M. Schmidt_, Dec 12 2013