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A237617 a(n) = n*(n + 1)*(17*n - 14)/6.

%I A237617 #35 Feb 16 2025 08:33:21
%S A237617 0,1,20,74,180,355,616,980,1464,2085,2860,3806,4940,6279,7840,9640,
%T A237617 11696,14025,16644,19570,22820,26411,30360,34684,39400,44525,50076,
%U A237617 56070,62524,69455,76880,84816,93280,102289,111860,122010,132756,144115,156104,168740
%N A237617 a(n) = n*(n + 1)*(17*n - 14)/6.
%C A237617 Also 19-gonal (or nonadecagonal) pyramidal numbers.
%C A237617 This sequence is related to A180232 by 2*a(n) = n*A180232(n) - Sum_{i=0..n-1} A180232(i).
%D A237617 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (seventeenth row of the table).
%H A237617 Bruno Berselli, <a href="/A237617/b237617.txt">Table of n, a(n) for n = 0..1000</a>
%H A237617 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>.
%H A237617 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H A237617 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A237617 G.f.: x*(1 + 16*x)/(1 - x)^4.
%F A237617 For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(17*i+1); see the generalization in A237616 (Formula field).
%F A237617 E.g.f.: x*(6 + 54*x + 17*x^2)*exp(x)/6. - _G. C. Greubel_, Aug 30 2019
%e A237617 After 0, the sequence is provided by the row sums of the triangle:
%e A237617    1;
%e A237617    2,  18;
%e A237617    3,  36,  35;
%e A237617    4,  54,  70,  52;
%e A237617    5,  72, 105, 104,  69;
%e A237617    6,  90, 140, 156, 138,  86;
%e A237617    7, 108, 175, 208, 207, 172, 103;
%e A237617    8, 126, 210, 260, 276, 258, 206, 120;
%e A237617    9, 144, 245, 312, 345, 344, 309, 240, 137;
%e A237617   10, 162, 280, 364, 414, 430, 412, 360, 274, 154; etc.,
%e A237617 where (r = row index, c = column index):
%e A237617 T(r,r) = T(c,c) = 17*r - 16 and T(r,c) = T(r-1,c) + T(r,r) = (r-c+1) * T(r,r), with r>=c>0.
%p A237617 seq(n*(n+1)*(17*n-14)/6, n=0..40); # _G. C. Greubel_, Aug 30 2019
%t A237617 Table[n(n+1)(17*n-14)/6, {n, 0, 40}]
%t A237617 CoefficientList[Series[x(1+16x)/(1-x)^4, {x,0,40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)
%t A237617 LinearRecurrence[{4,-6,4,-1},{0,1,20,74},40] (* _Harvey P. Dale_, Aug 04 2021 *)
%o A237617 (Magma) [n*(n+1)*(17*n-14)/6: n in [0..40]];
%o A237617 (Magma) I:=[0,1,20,74]; [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_, Feb 12 2014
%o A237617 (PARI) vector(40, n, n*(n-1)*(17*n-31)/6) \\ _G. C. Greubel_, Aug 30 2019
%o A237617 (Sage) [n*(n+1)*(17*n-14)/6 for n in (0..40)] # _G. C. Greubel_, Aug 30 2019
%o A237617 (GAP) List([0..40], n-> n*(n+1)*(17*n-14)/6); # _G. C. Greubel_, Aug 30 2019
%Y A237617 Cf. A051871, A180232.
%Y A237617 Cf. similar sequences listed in A237616.
%K A237617 nonn,easy
%O A237617 0,3
%A A237617 _Bruno Berselli_, Feb 11 2014