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%I A267370 #61 Sep 08 2022 08:46:15
%S A267370 0,6,21,48,90,150,231,336,468,630,825,1056,1326,1638,1995,2400,2856,
%T A267370 3366,3933,4560,5250,6006,6831,7728,8700,9750,10881,12096,13398,14790,
%U A267370 16275,17856,19536,21318,23205,25200,27306,29526,31863,34320,36900,39606,42441,45408,48510
%N A267370 Partial sums of A140091.
%C A267370 After 0, this sequence is the third column of the array in A185874.
%C A267370 Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.
%H A267370 Bruno Berselli, <a href="/A267370/b267370.txt">Table of n, a(n) for n = 0..1000</a>
%H A267370 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A267370 O.g.f.: 3*x*(2 - x)/(1 - x)^4.
%F A267370 E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
%F A267370 a(n) = n*(n + 1)*(n + 5)/2.
%F A267370 a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
%F A267370 a(n) = 3*A005581(n+1).
%F A267370 a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
%F A267370 From _Amiram Eldar_, Jan 06 2021: (Start)
%F A267370 Sum_{n>=1} 1/a(n) = 163/600.
%F A267370 Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)
%e A267370 The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
%e A267370 .  0;
%e A267370 .  1,  5;
%e A267370 .  4,  7, 10;
%e A267370 .  9, 11, 13, 15;
%e A267370 . 16, 17, 18, 19, 20;
%e A267370 . 25, 25, 25, 25, 25, 25;
%e A267370 . 36, 35, 34, 33, 32, 31, 30;
%e A267370 . 49, 47, 45, 43, 41, 39, 37, 35;
%e A267370 . 64, 61, 58, 55, 52, 49, 46, 43, 40;
%e A267370 . 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
%e A267370 First column is A000290.
%e A267370 Second column is A027690.
%e A267370 Third column is included in A189834.
%e A267370 Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
%e A267370 Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.
%t A267370 Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
%t A267370 LinearRecurrence[{4,-6,4,-1},{0,6,21,48},50] (* _Harvey P. Dale_, Jul 18 2019 *)
%o A267370 (PARI) vector(50, n, n--; n*(n+1)*(n+5)/2)
%o A267370 (Sage) [n*(n+1)*(n+5)/2 for n in (0..50)]
%o A267370 (Magma) [n*(n+1)*(n+5)/2: n in [0..50]];
%Y A267370 Cf. A005581, A051744, A105163, A140091, A185874.
%Y A267370 Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).
%K A267370 nonn,easy
%O A267370 0,2
%A A267370 _Bruno Berselli_, Jan 13 2016

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