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a(n) = n*(n + 1)*(19*n - 16)/6.
2

%I #32 May 28 2022 04:02:42

%S 0,1,22,82,200,395,686,1092,1632,2325,3190,4246,5512,7007,8750,10760,

%T 13056,15657,18582,21850,25480,29491,33902,38732,44000,49725,55926,

%U 62622,69832,77575,85870,94736,104192,114257,124950,136290,148296,160987,174382

%N a(n) = n*(n + 1)*(19*n - 16)/6.

%C Also 21-gonal (or icosihenagonal) pyramidal numbers.

%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (nineteenth row of the table).

%H Bruno Berselli, <a href="/A237618/b237618.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</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.: x*(1 + 18*x) / (1 - x)^4.

%F a(n) = (1/2)*( n*A226490(n) - Sum_{j=0..n-1} A226490(j) ).

%F a(n) = Sum_{i=0..n-1} (n-i)*(19*i+1), for n>0; see the generalization in A237616 (Formula field).

%F From _G. C. Greubel_, May 27 2022: (Start)

%F a(n) = binomial(n+2, 3) + 18*binomial(n+1, 3).

%F E.g.f.: (1/6)*x*(6 + 60*x + 19*x^2)*exp(x). (End)

%e After 0, the sequence is provided by the row sums of the triangle:

%e 1;

%e 2, 20;

%e 3, 40, 39;

%e 4, 60, 78, 58;

%e 5, 80, 117, 116, 77;

%e 6, 100, 156, 174, 154, 96;

%e 7, 120, 195, 232, 231, 192, 115;

%e 8, 140, 234, 290, 308, 288, 230, 134;

%e 9, 160, 273, 348, 385, 384, 345, 268, 153;

%e 10, 180, 312, 406, 462, 480, 460, 402, 306, 172; etc.,

%e where (r = row index, c = column index):

%e T(r,r) = T(c,c) = 19*r-18 and T(r,c) = T(r-1,c)+T(r,r) = (r-c+1)*T(r,r), with r>=c>0.

%t Table[n(n+1)(19n-16)/6, {n, 0, 40}]

%t CoefficientList[Series[x(1+18x)/(1-x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)

%o (Magma) [n*(n+1)*(19*n-16)/6: n in [0..40]];

%o (Magma) I:=[0,1,22,82]; [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..50]]; // _Vincenzo Librandi_, Feb 12 2014

%o (SageMath) b=binomial; [b(n+2,3) +18*b(n+1,3) for n in (0..50)] # _G. C. Greubel_, May 27 2022

%Y Cf. A051873, A226490.

%Y Cf. similar sequences listed in A237616.

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Feb 11 2014