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%I #77 Mar 11 2024 15:45:18
%S 0,1,10,51,180,501,1182,2471,4712,8361,14002,22363,34332,50973,73542,
%T 103503,142544,192593,255834,334723,432004,550725,694254,866295,
%U 1070904,1312505,1595906,1926315,2309356,2751085,3258006,3837087,4495776,5242017,6084266
%N Expansion of g.f. x*(1+x)^4/(1-x)^6.
%C Hyun Kwang Kim asserts that every nonnegative integer can be represented by the sum of no more than 14 of these numbers. - _Jonathan Vos Post_, Nov 16 2004
%C If Y_i (i=1,2,3,4) are 2-blocks of a (n+4)-set X then a(n-4) is the number of 9-subsets of X intersecting each Y_i (i=1,2,3,4). - _Milan Janjic_, Oct 28 2007
%C Starting with 1 = binomial transform of [1, 9, 32, 56, 48, 16, 0, 0, 0, ...], where (1, 9, 32, 56, 48, 16) = row 5 of the Chebyshev triangle A081277. Also = row 5 of the array in A142978. - _Gary W. Adamson_, Jul 19 2008
%C Starting with the term 1 this is the self-convolution of A001844(n). - _Anton Zakharov_, Sep 02 2016
%D H. S. M. Coxeter, Regular Polytopes, New York: Dover Publications, 1973.
%H Vincenzo Librandi, <a href="/A069038/b069038.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/enumfor.pdf">Two Enumerative Functions</a>
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F Recurrence: a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
%F a(n) = n*(2*n^4 + 10*n^2 + 3)/15. - _Jonathan Vos Post_, Nov 16 2004
%F a(n) = C(n+4,5) + 4*C(n+3,5) + 6*C(n+2,5) + 4*C(n+1,5) + C(n,5).
%F Sum_{n>=1} 1/((1/15)*n*(2*n^4 + 10*n^2 + 3)*n!) = hypergeom([1, 1, 1+i*sqrt(10-2*sqrt(19))*(1/2), 1-i*sqrt(10-2*sqrt(19))*(1/2), 1+i*sqrt(10+2*sqrt(19))*(1/2), 1-i*sqrt(10+2*sqrt(19))*(1/2)], [2, 2, 2+i*sqrt(10-2*sqrt(19))*(1/2), 2-i*sqrt(10-2*sqrt(19))*(1/2), 2+i*sqrt(10+2*sqrt(19))*(1/2), 2-i*sqrt(10+2*sqrt(19))*(1/2)], 1) = 1.05351734968093116819345664995829700099916... - _Stephen Crowley_, Jul 14 2009
%F a(n) = a(n-1) + A014820(n-1) + A014820(n-2). - _Bruce J. Nicholson_, Apr 18 2018
%F a(n) = 10*a(n-1)/(n-1) + a(n-2) for n > 1. - _Seiichi Manyama_, Jun 06 2018
%F Euler transform of length 2 sequence [10, -4]. - _Michael Somos_, Jun 19 2018
%F Sum_{k >= 1} (-1)^k/(a(k)*a(k+1)) = 10*log(2) - 41/6 = 1/(10 + 2/(10 + 6/(10 + ... + n*(n-1)/(10 + ...)))). See A142983. Cf. A005900 and A014820. - _Peter Bala_, Mar 08 2024
%F E.g.f.: exp(x)*x*(15 + 60*x + 60*x^2 + 20*x^3 + 2*x^4)/15. - _Stefano Spezia_, Mar 10 2024
%p al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(5,n),n=0..100)];
%t CoefficientList[Series[x (1 + x)^4/(1 - x)^6, {x, 0, 32}], x] (* _Michael De Vlieger_, Sep 02 2016 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,10,51,180,501},40] (* _Harvey P. Dale_, Jun 19 2021 *)
%o (Magma) [n*(2*n^4 + 10*n^2 + 3)/15: n in [0..40]]; // _Vincenzo Librandi_, May 22 2011
%o (PARI) concat(0, Vec(x*(1+x)^4/(1-x)^6 + O(x^99))) \\ _Altug Alkan_, Sep 02 2016
%o (PARI) {a(n) = n * (2*n^4 + 10*n^2 + 3) / 15}; /* _Michael Somos_, Jun 17 2018 */
%Y Cf. A000332, A014820, A005900.
%Y Cf. A142978, A081277.
%Y Cf. A001844.
%K nonn,easy
%O 0,3
%A _Vladeta Jovovic_, Apr 03 2002
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