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A050479 a(n) = C(n)*(9*n + 1) where C(n) = Catalan numbers (A000108).

%I A050479 #35 Aug 24 2025 11:30:00
%S A050479 1,10,38,140,518,1932,7260,27456,104390,398684,1528436,5878600,
%T A050479 22673308,87662200,339653880,1318498920,5126862150,19965297660,
%U A050479 77855108100,303969268680,1188105796020,4648590733800,18205030164360,71356399639200,279909199969308,1098799886728152
%N A050479 a(n) = C(n)*(9*n + 1) where C(n) = Catalan numbers (A000108).
%D A050479 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A050479 Andrew Howroyd, <a href="/A050479/b050479.txt">Table of n, a(n) for n = 0..200</a>
%F A050479 4*(n+1)*a(n) + (-23*n-1)*a(n-1) + 14*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Feb 13 2015
%F A050479 -(n+1)*(9*n-8)*a(n) + 2*(9*n+1)*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Feb 13 2015
%F A050479 G.f.: (4 - 7*x - 4*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - _Ilya Gutkovskiy_, Jun 13 2017
%F A050479 From _Peter Bala_, Aug 23 2025: (Start)
%F A050479 a(n) = binomial(2*n, n) + 8*binomial(2*n, n-1) = A000984(n) + 8*A001791(n).
%F A050479 a(n) ~ 4^n * 9/sqrt(Pi*n). (End)
%t A050479 A050479[n_] := CatalanNumber[n]*(9*n + 1);
%t A050479 Array[A050479, 30, 0] (* _Paolo Xausa_, Aug 24 2025 *)
%o A050479 (Magma) [Catalan(n)*(9*n+1):n in [0..27] ]; // _Marius A. Burtea_, Jan 05 2020
%o A050479 (Magma) R<x>:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (4-7*x-4*Sqrt(1-4*x))/(x*Sqrt(1-4*x))))); // _Marius A. Burtea_, Jan 05 2020
%Y A050479 Column k=9 of A330965.
%Y A050479 Cf. A017077, A000108, A051945.
%K A050479 easy,nonn,changed
%O A050479 0,2
%A A050479 _Barry E. Williams_, Dec 24 1999
%E A050479 Terms a(21) and beyond from _Andrew Howroyd_, Jan 05 2020