A256859 a(n) = n*(n + 1)*(n + 2)*(n^2 - n + 4)/24.

1, 6, 25, 80, 210, 476, 966, 1800, 3135, 5170, 8151, 12376, 18200, 26040, 36380, 49776, 66861, 88350, 115045, 147840, 187726, 235796, 293250, 361400, 441675, 535626, 644931, 771400, 916980, 1083760, 1273976, 1490016, 1734425, 2009910, 2319345, 2665776, 3052426

Offset: 1

Author

Luciano Ancora, Apr 14 2015

Keywords

• nonn
• easy

Comments

This is the case k = n of b(n,k) = n*(n+1)*(n+2)*(k*(n-1)+4)/24, where b(n,k) is the n-th hypersolid number in 4 dimensions generated from an arithmetical progression with the first term 1 and common difference k. Therefore, the sequence is the main diagonal of the Table 3 in Sardelis et al. paper (see Links field).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070v1 [math.GM], 2008.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000389, A261721.

Cf. similar sequences of the form binomial(n+k-2,k-1)+n*binomial(n+k-2,k): A006000 (k=2); A257055 (k=3); this sequence (k=4); A256860 (k=5); A256861 (k=6).

Programs

Magma

[n*(n + 1)*(n + 2)*(n^2 - n + 4)/24: n in [1..30]]; // G. C. Greubel, Nov 23 2017

Mathematica

Table[n (n + 1) (n + 2) (n^2 - n + 4)/24, {n, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,6,25,80,210,476},40] (* Harvey P. Dale, Mar 19 2022 *)

PARI

vector(40, n, n*(n+1)*(n+2)*(n^2-n+4)/24) \\ Bruno Berselli, Apr 15 2015

Formula

G.f.: x*(1 + 4*x^2)/(1 - x)^6.

a(n) = 4*A000389(n+2) + A000389(n+4). - Bruno Berselli, Apr 15 2015

E.g.f.: (24*x + 48*x^2 + 40*x^3 + 12*x^4 + x^5)*exp(x)/24. - G. C. Greubel, Nov 23 2017

a(n) = A261721(n,n-1). - Alois P. Heinz, Apr 15 2020