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Tenth column of trinomial coefficients.
3

%I #30 Aug 18 2024 15:54:08

%S 5,50,266,1016,3139,8350,19855,43252,87802,168168,306735,536640,

%T 905658,1481108,2355962,3656360,5550755,8260934,12075184,17363896,

%U 24597925,34370050,47419905,64662780,87222720,116470380,154066125,202008896,262691396,338962184

%N Tenth column of trinomial coefficients.

%H G. C. Greubel, <a href="/A064054/b064054.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F a(n) = A027907(n+5, 9).

%F a(n) = binomial(n+5, 5)*(n^4+66*n^3+1307*n^2+8706*n+15120) /(9!/5!).

%F G.f.: (1+x-x^2)*(5-5*x+x^2)/(1-x)^10, numerator polynomial is N3(9, x)= 5+0*x-9*x^2+6*x^3-x^4 from array A063420.

%F a(n) = A111808(n+5,9) for n>3. - _Reinhard Zumkeller_, Aug 17 2005

%F a(n) = 5*binomial(n+5,5) + 20*binomial(n+5,6) + 21*binomial(n+5,7) + 8*binomial(n+5,8) + binomial(n+5,9) (see our comment in A026729). - _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 22 2012

%F a(n) = GegenbauerC(N, -n, -1/2) where N = 9 if 9<n else 2*n-9. - _Peter Luschny_, May 10 2016

%p A064054 := n -> GegenbauerC(`if`(9<n,9,2*n-9), -n, -1/2):

%p seq(simplify(A064054(n)), n=5..20); # _Peter Luschny_, May 10 2016

%t Table[GegenbauerC[9, -n, -1/2], {n,5,50}] (* _G. C. Greubel_, Feb 28 2017 *)

%o (PARI) for(n=0,25, print1(binomial(n+5,5)*(n^4 + 66*n^3 + 1307*n^2 + 8706*n + 15120) /(9!/5!), ", ")) \\ _G. C. Greubel_, Feb 28 2017

%Y A005716 (ninth column), A111808.

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Aug 29 2001