A257199 a(n) = n*(n+1)*(n+2)*(n^2+2*n+17)/120.

1, 5, 16, 41, 91, 182, 336, 582, 957, 1507, 2288, 3367, 4823, 6748, 9248, 12444, 16473, 21489, 27664, 35189, 44275, 55154, 68080, 83330, 101205, 122031, 146160, 173971, 205871, 242296, 283712, 330616, 383537, 443037, 509712, 584193, 667147, 759278, 861328, 974078

Offset: 1

Author

Luciano Ancora, Apr 18 2015

Keywords

• nonn
• easy

Comments

Antidiagonal sums of the array of pyramidal numbers shown in Table 2 of Sardelis and Valahas paper (see A261720).

This is the case j = 3 of (n^2 + (j-1)*n + (j+1)^2 + 1)*binomial(n+j-1, j)/((j+1)*(j+2)), where j is the space dimension: a(n) = (n^2+2*n+17)*binomial(n+2,3)/20.

The sequence is the binomial transform of (1, 4, 7, 7, 4, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

Links

• D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070 [math.GM], 2008.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A257200, A257201.

For another version of the array, see A080851.

Programs

Magma

[n*(n+1)*(n+2)*(n^2+2*n+17)/120: n in [1..40]]; // Vincenzo Librandi, Apr 18 2015

Mathematica

Table[n (n + 1) (n + 2) (n^2 + 2n + 17)/120, {n, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,5,16,41,91,182},40] (* Harvey P. Dale, Mar 18 2018 *)

Formula

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