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%I #27 Feb 15 2022 08:16:18
%S 0,1,16,70,200,455,896,1596,2640,4125,6160,8866,12376,16835,22400,
%T 29240,37536,47481,59280,73150,89320,108031,129536,154100,182000,
%U 213525,248976,288666,332920,382075,436480,496496,562496,634865,714000,800310,894216,996151
%N Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).
%C Partial sums of A172073.
%C Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.
%H Bruno Berselli, <a href="/A220212/b220212.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.
%F G.f.: x*(1+11*x)/(1-x)^5.
%F a(n) = n*(n+1)*(n+2)*(3*n-2)/6.
%F From _Amiram Eldar_, Feb 15 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)
%t A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
%t CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%o (Magma) A051866:=func<n | n*(6*n-5)>; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];
%o (Magma) I:=[0,1,16,70,200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // _Vincenzo Librandi_, Aug 18 2013
%Y Cf. A135021, A172073.
%Y Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):
%Y k= 2 (A000027 ): A000292;
%Y k= 3 (A000217 ): A000332 (after the third term);
%Y k= 4 (A000290 ): A002415 (after the first term);
%Y k= 5 (A000326 ): A001296;
%Y k= 6 (A000384*): A002417;
%Y k= 7 (A000566 ): A002418;
%Y k= 8 (A000567*): A002419;
%Y k= 9 (A001106*): A051740;
%Y k=10 (A001107*): A051797;
%Y k=11 (A051682*): A051798;
%Y k=12 (A051624*): A051799;
%Y k=13 (A051865*): A055268.
%Y Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.
%K nonn,easy
%O 0,3
%A _Bruno Berselli_, Dec 08 2012
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