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%I #51 Sep 08 2022 08:46:07
%S 0,5,23,65,145,280,490,798,1230,1815,2585,3575,4823,6370,8260,10540,
%T 13260,16473,20235,24605,29645,35420,41998,49450,57850,67275,77805,
%U 89523,102515,116870,132680,150040,169048,189805,212415,236985,263625,292448
%N a(n) = n*(n + 1)*(n + 2)*(3*n + 17)/24.
%C Equivalently, Sum_{i=0..n} (i+4)*A000217(i).
%C Sequences of the type Sum_{i=0..n} (i+k)*A000217(i):
%C k = 0, A001296: 0, 1, 7, 25, 65, 140, 266, 462, ...
%C k = 1, A000914: 0, 2, 11, 35, 85, 175, 322, 546, ...
%C k = 2, A050534: 0, 3, 15, 45, 105, 210, 378, 630, ... (deleting two 0)
%C k = 3, A215862: 0, 4, 19, 55, 125, 245, 434, 714, ...
%C k = 4, a(n): 0, 5, 23, 65, 145, 280, 490, 798, ...
%C k = 5, A239568: 0, 6, 27, 75, 165, 315, 546, 882, ...
%C Antidiagonal sums (without 0) give A034263: 1, 9, 39, 119, 294, ...
%C Diagonal: 1, 11, 45, 125, 280, 546, ... is A051740.
%C Also: k = -1 gives A050534 deleting a 0; k = -2 gives 0 followed by A059302.
%C After 0, partial sums of A212343 and third column of A118788.
%C This sequence is even related to A005286 by a(n) = n*A005286(n) - Sum_{i=0..n-1} A005286(i).
%H Bruno Berselli, <a href="/A241765/b241765.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).
%F G.f.: x*(5 - 2*x)/(1 - x)^5.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F a(n) = A227342(A055998(n+1)).
%F a(n) = Sum_{j=0..n+2} (-1)^(n-j)*binomial(-j,-n-2)*S1(j,n), S1 Stirling cycle numbers A132393. - _Peter Luschny_, Apr 10 2016
%e a(7) = 4*0 + 5*1 + 6*3 + 7*6 + 8*10 + 9*15 + 10*21 + 11*28 = 798.
%p A241765:=n->n*(n + 1)*(n + 2)*(3*n + 17)/24; seq(A241765(n), n=0..40); # _Wesley Ivan Hurt_, May 09 2014
%t Table[n (n + 1) (n + 2) (3 n + 17)/24, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 23, 65, 145}, 40]
%t CoefficientList[Series[x (5 - 2 x)/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 09 2014 *)
%o (Sage) [n*(n+1)*(n+2)*(3*n+17)/24 for n in (0..40)]
%o (Maxima) makelist(coeff(taylor(x*(5-2*x)/(1-x)^5, x, 0, n), x, n), n, 0, 40);
%o (Magma) /* By first comment: */ k:=4; A000217:=func<i | i*(i+1)/2>; [&+[(i+k)*A000217(i): i in [0..n]]: n in [0..40]];
%o (PARI) a(n)=n*(n+1)*(n+2)*(3*n+17)/24 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (PARI) x='x+O('x^99); concat(0, Vec(x*(5-2*x)/(1-x)^5)) \\ _Altug Alkan_, Apr 10 2016
%Y Cf. A000217, A005286, A118788, A212343, A227342.
%Y Cf. similar sequences A000914, A001296, A050534, A059302, A215862, A239568 (see table in Comments lines).
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Apr 28 2014
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