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%I #48 Sep 08 2022 08:44:35
%S 1,6,21,56,126,251,456,771,1231,1876,2751,3906,5396,7281,9626,12501,
%T 15981,20146,25081,30876,37626,45431,54396,64631,76251,89376,104131,
%U 120646,139056,159501,182126
%N 4-dimensional centered tetrahedral numbers.
%C Binomial transform of (1,5,10,10,5,0,0,0,...). - _Paul Barry_, Jul 01 2003
%C If X is an n-set and Y a fixed 5-subset of X then a(n-5) is equal to the number of 5-subsets of X intersecting Y. - _Milan Janjic_, Jul 30 2007
%C Also the sum of five consecutive terms of A000332. - _Bruno Berselli_, Jun 18 2015
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 224 (general formula for n-th centered polytope number).
%H Vincenzo Librandi, <a href="/A008498/b008498.txt">Table of n, a(n) for n = 0..10000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (1-x^5)/(1-x)^6 = (1 +x +x^2 +x^3 +x^4)/(1-x)^5.
%F a(n) = C(n,0) + 5*C(n,1) + 10*C(n,2) + 10*C(n,3) + 5*C(n,4). - _Paul Barry_, Jul 01 2003
%F a(n) = (5*n^4 + 10*n^3 + 55*n^2 + 50*n + 24)/24. - _Paul Barry_, Jul 01 2003
%F a(n) = binomial(n+5,5) - binomial(n,5). - _Zerinvary Lajos_, Jul 21 2006
%F a(n) = 1 + 5*A006522(n+2). - _Bruno Berselli_, Jun 18 2015
%F E.g.f.: (24 + 120*x + 120*x^2 + 40*x^3 + 5*x^4)*exp(x)/24. - _G. C. Greubel_, Nov 08 2019
%p [seq(binomial(n+5,5)-binomial(n,5), n=0..45)]; # _Zerinvary Lajos_, Jul 21 2006
%t LinearRecurrence[{5,-10,10,-5,1}, {1,6,21,56,126}, 40] (* _Harvey P. Dale_, Dec 18 2013 *)
%t Table[1 + 5n(n+1)(n^2 +n +10)/24, {n, 0, 40}] (* _Bruno Berselli_, Jun 18 2015 *)
%o (Magma) [(5*n^4+10*n^3+55*n^2+50*n+24)/24: n in [0..30] ]; // _Vincenzo Librandi_, Aug 21 2011
%o (Magma) [Binomial(n+5,5) - Binomial(n,5): n in [0..40]]; // _G. C. Greubel_, Nov 08 2019
%o (Sage) [binomial(n+5,5) - binomial(n,5) for n in (0..40)] # _G. C. Greubel_, Nov 08 2019
%o (GAP) List([0..40], n-> Binomial(n+5,5) - Binomial(n,5)); # _G. C. Greubel_, Nov 08 2019
%Y Cf. A000332, A005894, A006522.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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