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A237618 a(n) = n*(n + 1)*(19*n - 16)/6.

%I A237618 #33 Feb 16 2025 08:33:21
%S A237618 0,1,22,82,200,395,686,1092,1632,2325,3190,4246,5512,7007,8750,10760,
%T A237618 13056,15657,18582,21850,25480,29491,33902,38732,44000,49725,55926,
%U A237618 62622,69832,77575,85870,94736,104192,114257,124950,136290,148296,160987,174382
%N A237618 a(n) = n*(n + 1)*(19*n - 16)/6.
%C A237618 Also 21-gonal (or icosihenagonal) pyramidal numbers.
%D A237618 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (nineteenth row of the table).
%H A237618 Bruno Berselli, <a href="/A237618/b237618.txt">Table of n, a(n) for n = 0..1000</a>
%H A237618 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>.
%H A237618 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A237618 G.f.: x*(1 + 18*x) / (1 - x)^4.
%F A237618 a(n) = (1/2)*( n*A226490(n) - Sum_{j=0..n-1} A226490(j) ).
%F A237618 a(n) = Sum_{i=0..n-1} (n-i)*(19*i+1), for n>0; see the generalization in A237616 (Formula field).
%F A237618 From _G. C. Greubel_, May 27 2022: (Start)
%F A237618 a(n) = binomial(n+2, 3) + 18*binomial(n+1, 3).
%F A237618 E.g.f.: (1/6)*x*(6 + 60*x + 19*x^2)*exp(x). (End)
%e A237618 After 0, the sequence is provided by the row sums of the triangle:
%e A237618    1;
%e A237618    2,  20;
%e A237618    3,  40,  39;
%e A237618    4,  60,  78,  58;
%e A237618    5,  80, 117, 116, 77;
%e A237618    6, 100, 156, 174, 154, 96;
%e A237618    7, 120, 195, 232, 231, 192, 115;
%e A237618    8, 140, 234, 290, 308, 288, 230, 134;
%e A237618    9, 160, 273, 348, 385, 384, 345, 268, 153;
%e A237618   10, 180, 312, 406, 462, 480, 460, 402, 306, 172; etc.,
%e A237618 where (r = row index, c = column index):
%e A237618 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 A237618 Table[n(n+1)(19n-16)/6, {n, 0, 40}]
%t A237618 CoefficientList[Series[x(1+18x)/(1-x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)
%o A237618 (Magma) [n*(n+1)*(19*n-16)/6: n in [0..40]];
%o A237618 (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 A237618 (SageMath) b=binomial; [b(n+2,3) +18*b(n+1,3) for n in (0..50)] # _G. C. Greubel_, May 27 2022
%Y A237618 Cf. A051873, A226490.
%Y A237618 Cf. similar sequences listed in A237616.
%K A237618 nonn,easy
%O A237618 0,3
%A A237618 _Bruno Berselli_, Feb 11 2014