This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A101853 #31 Oct 15 2022 02:07:30
%S A101853 6,18,37,64,100,146,203,272,354,450,561,688,832,994,1175,1376,1598,
%T A101853 1842,2109,2400,2716,3058,3427,3824,4250,4706,5193,5712,6264,6850,
%U A101853 7471,8128,8822,9554,10325,11136,11988,12882,13819,14800
%N A101853 a(n) = n*(20 + 15*n + n^2)/6.
%C A101853 4th partial summation within series as series accumulate n times from an initial sequence of Euler Triangle's row 3: 1,4,1. The 1,4,1 is the left column, A101101 the second column, A008458 the third, A003215 the fourth column etc of the array in the example. a(n) is the 4th row.
%H A101853 Vincenzo Librandi, <a href="/A101853/b101853.txt">Table of n, a(n) for n = 1..1000</a>
%H A101853 C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>.
%H A101853 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A101853 G.f.: x*(6 - 6*x + x^2)/(x - 1)^4. - _R. J. Mathar_, Dec 06 2011
%F A101853 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 26 2012
%F A101853 E.g.f.: exp(x)*x*(36 + 18*x + x^2)/6. - _Stefano Spezia_, Oct 14 2022
%e A101853 Left column the third row of A008292, and subsequent columns defined as partial sums along their preceding neighbors:
%e A101853 1 1 1 1 1 1 1 1 1 1 1
%e A101853 4 5 6 7 8 9 10 11 12 13 14
%e A101853 1 6 12 19 27 36 46 57 69 82 96 A051936
%e A101853 0 6 18 37 64 100 146 203 272 354 450 A101853
%e A101853 0 6 24 61 125 225 371 574 846 1200 1650 A101854
%e A101853 0 6 30 91 216 441 812 1386 2232 3432 5082 A101855
%e A101853 0 6 36 127 343 784 1596 2982 5214 8646 13728
%e A101853 0 6 42 169 512 1296 2892 5874 11088 19734 33462
%e A101853 0 6 48 217 729 2025 4917 10791 21879 41613 75075
%e A101853 ...
%t A101853 LinearRecurrence[{4,-6,4,-1},{6,18,37,64},40] (* or *) CoefficientList[Series[(6-6*x+x^2)/(x-1)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 26 2012 *)
%o A101853 (Magma) I:=[6, 18, 37, 64]; [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..40]]; // _Vincenzo Librandi_, Jun 26 2012
%o A101853 (PARI) a(n)=n*(20+15*n+n^2)/6 \\ _Charles R Greathouse IV_, Oct 16 2015
%Y A101853 Row n=3 of A255961.
%Y A101853 Cf. A003215, A008292, A008458, A101101.
%K A101853 easy,nonn
%O A101853 1,1
%A A101853 Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 18 2004