%I #17 Oct 24 2025 06:15:58

%S 1,30,175,588,1485,3146,5915,10200,16473,25270,37191,52900,73125,

%T 98658,130355,169136,215985,271950,338143,415740,505981,610170,729675,

%U 865928,1020425,1194726,1390455,1609300,1853013,2123410,2422371,2751840,3113825,3510398,3943695

%N Bisection of A002417.

%H G. C. Greubel, <a href="/A100430/b100430.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = (8*n^4 +28*n^3 +34*n^2 +17*n+3)/3. - _Ralf Stephan_, May 15 2007

%F From _G. C. Greubel_, Apr 09 2023: (Start)

%F a(n) = (2*n+1)*binomial(2*n+3, 3).

%F a(n) = (2*n+1)*A000447(n+1).

%F G.f.: (1 + 25*x + 35*x^2 + 3*x^3)/(1-x)^5.

%F E.g.f.: (1/3)*(3 + 87*x + 174*x^2 + 76*x^3 + 8*x^4)*exp(x). (End)

%F From _Amiram Eldar_, Oct 24 2025: (Start)

%F Sum_{n>=0} 1/a(n) = 3*Pi^2/8 - 6*log(2) + 3/2.

%F Sum_{n>=0} (-1)^n/a(n) = 3*G - 3*Pi/4 + 3*log(2) - 3/2, where G is Catalan's constant (A006752). (End)

%t Table[(2*n+1)*Binomial[2*n+3,3], {n,0,50}] (* _G. C. Greubel_, Apr 09 2023 *)

%o (Magma) [(2*n+1)*Binomial(2*n+3,3): n in [0..50]]; // _G. C. Greubel_, Apr 09 2023

%o (SageMath) [(2*n+1)*binomial(2*n+3,3) for n in range(51)] # _G. C. Greubel_, Apr 09 2023

%Y Cf. A000447, A002417, A006752, A100431.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Nov 20 2004

%E More terms from _Hugo Pfoertner_, Nov 26 2004