%I #23 Feb 08 2022 22:13:15
%S 1,7,29,92,246,582,1254,2508,4719,8437,14443,23816,38012,58956,89148,
%T 131784,190893,271491,379753,523204,710930,953810,1264770,1659060,
%U 2154555,2772081,3535767,4473424,5616952,7002776,8672312,10672464,13056153,15882879,19219317,23139948,27727726,33074782,39283166,46465628
%N a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n^2+4*n+37)/5040.
%C Antidiagonal sums of the array of 5-dimensional solid numbers (see Example field).
%C See A257199 (second comment) for the general formula of this type of numbers: the sequence correspond to the case j = 5.
%C The sequence is the binomial transform of (1, 6, 16, 25, 25, 16, 6, 1, 0, 0, 0, ...). - _Gary W. Adamson_, Aug 26 2015
%H D. A. Sardelis and T. M. Valahas, <a href="http://arxiv.org/abs/0805.4070">On Multidimensional Pythagorean Numbers</a>, arXiv:0805.4070 [math.GM], 2008.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F G.f.: x*(1 - x + x^2)/(1 - x)^8.
%e Array in Comments begins:
%e 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, ...
%e 1, 7, 27, 77, 182, 378, 714, 1254, 2079, 3289, ...
%e 1, 8, 33, 98, 238, 504, 966, 1716, 2871, 4576, ...
%e 1, 9, 39, 119, 294, 630, 1218, 2178, 3663, 5863, ...
%e 1, 10, 45, 140, 350, 756, 1470, 2640, 4455, 7150, ...
%e 1, 11, 51, 161, 406, 882, 1722, 3102, 5247, 8437, ...
%e 1, 12, 57, 182, 462, 1008, 1974, 3564, 6039, 9724, ...
%e ...
%t Table[n (n + 1) (n + 2) (n + 3) (n + 4) (n^2 + 4n + 37)/5040, {n, 40}]
%o (Magma) [n*(n+1)*(n+2)*(n+3)*(n+4)*(n^2+4*n+37)/5040: n in [1..40]]; // _Vincenzo Librandi_, Apr 18 2015
%Y Cf. A257199, A257200.
%K nonn,easy
%O 1,2
%A _Luciano Ancora_, Apr 18 2015