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A257201 a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n^2+4*n+37)/5040.

%I A257201 #23 Feb 08 2022 22:13:15
%S A257201 1,7,29,92,246,582,1254,2508,4719,8437,14443,23816,38012,58956,89148,
%T A257201 131784,190893,271491,379753,523204,710930,953810,1264770,1659060,
%U A257201 2154555,2772081,3535767,4473424,5616952,7002776,8672312,10672464,13056153,15882879,19219317,23139948,27727726,33074782,39283166,46465628
%N A257201 a(n) =  n*(n+1)*(n+2)*(n+3)*(n+4)*(n^2+4*n+37)/5040.
%C A257201 Antidiagonal sums of the array of 5-dimensional solid numbers (see Example field).
%C A257201 See A257199 (second comment) for the general formula of this type of numbers: the sequence correspond to the case j = 5.
%C A257201 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 A257201 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 A257201 <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 A257201 G.f.: x*(1 - x + x^2)/(1 - x)^8.
%e A257201 Array in Comments begins:
%e A257201   1,  6, 21,  56, 126,  252,  462,  792, 1287, 2002, ...
%e A257201   1,  7, 27,  77, 182,  378,  714, 1254, 2079, 3289, ...
%e A257201   1,  8, 33,  98, 238,  504,  966, 1716, 2871, 4576, ...
%e A257201   1,  9, 39, 119, 294,  630, 1218, 2178, 3663, 5863, ...
%e A257201   1, 10, 45, 140, 350,  756, 1470, 2640, 4455, 7150, ...
%e A257201   1, 11, 51, 161, 406,  882, 1722, 3102, 5247, 8437, ...
%e A257201   1, 12, 57, 182, 462, 1008, 1974, 3564, 6039, 9724, ...
%e A257201   ...
%t A257201 Table[n (n + 1) (n + 2) (n + 3) (n + 4) (n^2 + 4n + 37)/5040, {n, 40}]
%o A257201 (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 A257201 Cf. A257199, A257200.
%K A257201 nonn,easy
%O A257201 1,2
%A A257201 _Luciano Ancora_, Apr 18 2015