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a(n) = T(2n+1, n+4), T given by A027948.
1

%I #11 Sep 08 2022 08:44:49

%S 1,9,175,1518,8735,39130,148487,502415,1568062,4622488,13091798,

%T 36067176,97522270,260459265,690141333,1819657318,4783398669,

%U 12551942930,32903246829,86201363911,225761428636,591165917888,1547848480940,4052529200192,10609936578716,27777538280521

%N a(n) = T(2n+1, n+4), T given by A027948.

%H G. C. Greubel, <a href="/A027956/b027956.txt">Table of n, a(n) for n = 3..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (11,-53,148,-266,322,-266,148,-53,11,-1).

%F From _G. C. Greubel_, Sep 30 2019: (Start)

%F a(n) = Sum_{j=0..n-3} binomial(2*n-j+1, j+8) for n >= 4.

%F a(n) = Fibonacci(2*n+10) - (34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630 for n >= 4.

%F G.f.: x^3*(1 -2*x +129*x^2 -78*x^3 +246*x^4 -329*x^5 +266*x^6 -148*x^7 +53*x^8 -11*x^9 +x^10)/((1-x)^8*(1-3*x+x^2)). (End)

%p with(combinat); seq(`if`(n=3,1, fibonacci(2*n+10) -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630), n=3..40); # _G. C. Greubel_, Sep 30 2019

%t Table[If[n==3, 1, Fibonacci[2*n+10] -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630], {n, 3, 40}] (* _G. C. Greubel_, Sep 30 2019 *)

%t LinearRecurrence[{11,-53,148,-266,322,-266,148,-53,11,-1},{1,9,175,1518,8735,39130,148487,502415,1568062,4622488,13091798},40] (* _Harvey P. Dale_, Sep 12 2021 *)

%o (PARI) vector(40, n, my(m=n+2); if(m==3, 1, fibonacci(2*m+10) -(34650 +33360*m +16065*m^2 +5089*m^3 +1260*m^4 +280*m^5 +16*m^7)/630) ) \\ _G. C. Greubel_, Sep 30 2019

%o (Magma) [1] cat [Fibonacci(2*n+10) -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630: n in [4..40]]; // _G. C. Greubel_, Sep 30 2019

%o (Sage) [1]+[fibonacci(2*n+10) -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630 for n in (4..40)] # _G. C. Greubel_, Sep 30 2019

%o (GAP) Concatenation([1], List([4..40], n-> Fibonacci(2*n+10) -(34650 +33360*n +16065*n^2 +5089*n^3 +1260*n^4 +280*n^5 +16*n^7)/630) ); # _G. C. Greubel_, Sep 30 2019

%Y Cf. A000045, A027948.

%K nonn

%O 3,2

%A _Clark Kimberling_

%E Name corrected and terms a(22) onward by _G. C. Greubel_, Sep 30 2019