VOOZH about

URL: https://oeis.org/A241765

⇱ A241765 - OEIS

login
A241765
a(n) = n*(n + 1)*(n + 2)*(3*n + 17)/24.
12
0, 5, 23, 65, 145, 280, 490, 798, 1230, 1815, 2585, 3575, 4823, 6370, 8260, 10540, 13260, 16473, 20235, 24605, 29645, 35420, 41998, 49450, 57850, 67275, 77805, 89523, 102515, 116870, 132680, 150040, 169048, 189805, 212415, 236985, 263625, 292448, 323570, 357110
OFFSET
0,2
COMMENTS
Equivalently, a(n) = Sum_{i=0..n} (i+4)*A000217(i).
Sequences of the type Sum_{i=0..n} (i+k)*A000217(i):
k = 0, A001296: 0, 1, 7, 25, 65, 140, 266, 462, ...
k = 1, A000914: 0, 2, 11, 35, 85, 175, 322, 546, ...
k = 2, A050534: 0, 3, 15, 45, 105, 210, 378, 630, ... (deleting two 0)
k = 3, A215862: 0, 4, 19, 55, 125, 245, 434, 714, ...
k = 4, a(n): 0, 5, 23, 65, 145, 280, 490, 798, ...
k = 5, A239568: 0, 6, 27, 75, 165, 315, 546, 882, ...
Antidiagonal sums (without 0) give A034263: 1, 9, 39, 119, 294, ...
Diagonal: 1, 11, 45, 125, 280, 546, ... is A051740.
Also: k = -1 gives A050534 deleting a 0; k = -2 gives 0 followed by A059302.
After 0, partial sums of A212343 and third column of A118788.
This sequence is even related to A005286 by a(n) = n*A005286(n) - Sum_{i=0..n-1} A005286(i).
FORMULA
G.f.: x*(5 - 2*x)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A227342(A055998(n+1)).
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
From Amiram Eldar, Oct 25 2025: (Start)
Sum_{n>=1} 1/a(n) = 5772117/17134810 + 18*sqrt(3)*Pi/1309 - 162*log(3)/1309.
Sum_{n>=1} (-1)^(n+1)/a(n) = 672*log(2)/187 - 37249827/17134810 - 36*sqrt(3)*Pi/1309. (End)
EXAMPLE
a(7) = 4*0 + 5*1 + 6*3 + 7*6 + 8*10 + 9*15 + 10*21 + 11*28 = 798.
MAPLE
A241765:=n->n*(n + 1)*(n + 2)*(3*n + 17)/24; seq(A241765(n), n=0..40); # Wesley Ivan Hurt, May 09 2014
MATHEMATICA
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]
CoefficientList[Series[x (5 - 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)
PROG
(SageMath) [n*(n+1)*(n+2)*(3*n+17)/24 for n in (0..40)]
(Maxima) makelist(coeff(taylor(x*(5-2*x)/(1-x)^5, x, 0, n), x, n), n, 0, 40);
(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]];
(PARI) a(n)=n*(n+1)*(n+2)*(3*n+17)/24 \\ Charles R Greathouse IV, Oct 07 2015
(PARI) my(x='x+O('x^99)); concat(0, Vec(x*(5-2*x)/(1-x)^5)) \\ Altug Alkan, Apr 10 2016
CROSSREFS
Cf. similar sequences A000914, A001296, A050534, A059302, A215862, A239568 (see table in Comments lines).
Sequence in context: A373538 A176874 A331987 * A106956 A084671 A284648
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Apr 28 2014
STATUS
approved