A210569 a(n) = (n-3)*(n-2)*(n-1)*n*(n+1)/30.

0, 0, 0, 0, 4, 24, 84, 224, 504, 1008, 1848, 3168, 5148, 8008, 12012, 17472, 24752, 34272, 46512, 62016, 81396, 105336, 134596, 170016, 212520, 263120, 322920, 393120, 475020, 570024, 679644, 805504, 949344, 1113024, 1298528, 1507968, 1743588, 2007768, 2303028

Offset: 0

Author

Bruno Berselli, Mar 23 2012

Keywords

• nonn
• easy

Comments

The following sequences are provided by the formula n*binomial(n,k) - binomial(n,k+1) = k*binomial(n+1,k+1):

. A000217(n) for k=1,

. A007290(n+1) for k=2,

. A050534(n) for k=3,

. a(n) for k=4,

. A000910(n+1) for k=5.

Sum of reciprocals of a(n), for n>3: 5/16.

From a(2), convolution of oblong numbers (A002378) with themselves. - Bruno Berselli, Oct 24 2016

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 3. - _N. J. A. Sloane_, Mar 23 2014
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000217, A000389, A000910, A002378, A007290, A050534, A177747.

First differences are in A033488.

Programs

Magma

[4*Binomial(n+1,5): n in [0..38]];

Maple

f:=n->(n^5-5*n^3+4*n)/30;
[seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 23 2014

Mathematica

LinearRecurrence[{6,-15,20,-15,6,-1}, {0,0,0,0,4,24}, 39]
CoefficientList[Series[4x^4/(1-x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Times@@@Partition[Range[-3,40],5,1]/30 (* Harvey P. Dale, Sep 19 2020 *)

Maxima

makelist(coeff(taylor(4*x^4/(1-x)^6, x, 0, n), x, n), n, 0, 38);

PARI

a(n)=(n-3)*(n-2)*(n-1)*n*(n+1)/30 \\ Charles R Greathouse IV, Oct 07 2015

SageMath

[4*binomial(n+1,5) for n in (0..40)] # G. C. Greubel, May 23 2022

Formula

G.f.: 4*x^4/(1-x)^6.

a(n) = n*binomial(n,4)-binomial(n,5) = 4*binomial(n+1,5) = 4*A000389(n+1).

a(n) = 2*A177747(2*n-7), with A177747(-7) = A177747(-5) = A177747(-3) = A177747(-1) = 0.

(n-4)*a(n) = (n+1)*a(n-1).

E.g.f.: (1/30)*x^4*(5+x)*exp(x). - G. C. Greubel, May 23 2022

Sum_{n>=4} (-1)^n/a(n) = 20*log(2) - 655/48. - Amiram Eldar, Jun 02 2022