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a(n) = (3+n)*(2 + 33*n + n^2)/6.
3

%I #24 Sep 08 2022 08:45:16

%S 1,24,60,110,175,256,354,470,605,760,936,1134,1355,1600,1870,2166,

%T 2489,2840,3220,3630,4071,4544,5050,5590,6165,6776,7424,8110,8835,

%U 9600,10406,11254,12145,13080,14060,15086,16159,17280,18450,19670,20941

%N a(n) = (3+n)*(2 + 33*n + n^2)/6.

%C The 4th partial summation within series as series accumulate n times from an initial sequence of Euler Triangle's row 4: 1,11,11,1.

%C The partial sums of A101859 (plus a leading 1). 4th row in the array shown in the examples. The 2nd column is A101104, the 3rd column is A101103, the 4th column is A005914.

%H Vincenzo Librandi, <a href="/A101860/b101860.txt">Table of n, a(n) for n = 0..1000</a>

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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: ( 1 + 20*x - 30*x^2 + 10*x^3 ) / (x-1)^4 . - _R. J. Mathar_, Dec 06 2011

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 26 2012

%e Array with first column equal to the 4th row of A008292, and column k defined by partial sums of the preceding column k-1:

%e 1 1 1 1 1 1 1 1 1 1 1

%e 11 12 13 14 15 16 17 18 19 20 21

%e 11 23 36 50 65 81 98 116 135 155 176

%e 1 24 60 110 175 256 354 470 605 760 936 A101860

%e 0 24 84 194 369 625 979 1449 2054 2814 3750 A101861

%e 0 24 108 302 671 1296 2275 3724 5778 8592 12342 A101862

%e 0 24 132 434 1105 2401 4676 8400 14178 22770 35112

%e 0 24 156 590 1695 4096 8772 17172 31350 54120 89232

%e 0 24 180 770 2465 6561 15333 32505 63855 117975 207207

%e ... ... ... ... ... ... ... ... ... ...

%t LinearRecurrence[{4,-6,4,-1},{1,24,60,110},50] (* or *) CoefficientList[Series[(1+20*x-30*x^2+10*x^3)/(x-1)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Jun 26 2012 *)

%o (Magma) I:=[1, 24, 60, 110]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // _Vincenzo Librandi_, Jun 26 2012

%o (PARI) a(n) = (n+3)*(n^2+33*n+2)/6; \\ _Altug Alkan_, Sep 23 2018

%K nonn,easy

%O 0,2

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