A103604 - OEIS

%I #26 Jun 30 2026 21:36:51

%S 210,3234,25872,144144,630630,2312310,7399392,21237216,55747692,

%T 135795660,310390080,671571264,1385115732,2738894004,5216940960,

%U 9610154400,17178150990,29881321470,50707697040,84126042000,136704818250,217946538810,341398774080,526116951360

%N a(n) = binomial(n+6,6) * binomial(n+10,6).

%H T. D. Noe, <a href="/A103604/b103604.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

%F G.f.: 42*(5+12*x+5*x^2) / (1-x)^13. - _Colin Barker_, Jul 01 2015

%F a(n) = A000579(n+6)*A000579(n+10). - _Michel Marcus_, Jul 01 2015

%F From _Amiram Eldar_, Sep 06 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 60*Pi^2 - 10445899/17640.

%F Sum_{n>=0} (-1)^n/a(n) = 447173/2205 - 2048*log(2)/7. (End)

%t Table[Binomial[n+6,6]Binomial[n+10,6],{n,0,30}] (* _Harvey P. Dale_, Apr 18 2019 *)

%t (* Alternative: *)

%t LinearRecurrence[ {13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{210,3234,25872, 144144,630630,2312310,7399392,21237216,55747692,135795660,310390080, 671571264,1385115732},30] (* _Harvey P. Dale_, Apr 18 2019 *)

%o (PARI) a(n) = binomial(n+6,6)*binomial(n+10,6) \\ _Colin Barker_, Jul 01 2015

%o (PARI) Vec(-42*(5*x^2+12*x+5)/(x-1)^13 + O(x^30)) \\ _Colin Barker_, Jul 01 2015

%o (Magma)

%o A103604:= func< n | Binomial(n+6,6)*Binomial(n+10,6) >;

%o [A103604(n): n in [0..30]]; // _G. C. Greubel_, Mar 05 2025

%o (SageMath)

%o def A103604(n): return binomial(n+6,6)*binomial(n+10,6)

%o print([A103604(n) for n in range(31)]) # _G. C. Greubel_, Mar 05 2025

%Y Cf. A000579, A062264.

%K nonn,easy,changed

%O 0,1

%A _Zerinvary Lajos_, Apr 22 2005