A267370 Partial sums of A140091.

0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510

Offset

0,2

Comments

After 0, this sequence is the third column of the array in A185874.

Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.

Links

Bruno Berselli, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

O.g.f.: 3*x*(2 - x)/(1 - x)^4.

E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.

a(n) = n*(n + 1)*(n + 5)/2.

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.

a(n) = 3*A005581(n+1).

a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.

From Amiram Eldar, Jan 06 2021: (Start)

Sum_{n>=1} 1/a(n) = 163/600.

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)

Example

The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
.  0;
.  1,  5;
.  4,  7, 10;
.  9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.

Mathematica

Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4, -6, 4, -1}, {0, 6, 21, 48}, 50] (* Harvey P. Dale, Jul 18 2019 *)

Prog

(PARI) vector(50, n, n--; n*(n+1)*(n+5)/2)
(Sage) [n*(n+1)*(n+5)/2 for n in (0..50)]
(Magma) [n*(n+1)*(n+5)/2: n in [0..50]];

Crossrefs

Cf. A005581, A051744, A105163, A140091, A185874.

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).

Sequence in context: A212656 A051941 A212707 * A213388 A163715 A028345

Adjacent sequences: A267367 A267368 A267369 * A267371 A267372 A267373

Keyword

nonn,easy

Author

Bruno Berselli, Jan 13 2016

Status

approved