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A004331 Binomial coefficient C(4n,n-1).

%I A004331 #37 Mar 19 2025 08:44:36
%S A004331 1,8,66,560,4845,42504,376740,3365856,30260340,273438880,2481256778,
%T A004331 22595200368,206379406870,1889912732400,17345898649800,
%U A004331 159518999862720,1469568786235308,13559593014190944,125288932441604200
%N A004331 Binomial coefficient C(4n,n-1).
%D A004331 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%H A004331 Seiichi Manyama, <a href="/A004331/b004331.txt">Table of n, a(n) for n = 1..1000</a>
%H A004331 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A004331 G.f.: (g^2-g)/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - _Mark van Hoeij_, Nov 11 2011
%F A004331 With an offset of 0, the o.g.f. equals f(x)*g(x)^4, where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A052203 (k = 1), A257633 (k = 2) and A224274 (k = 3). - _Peter Bala_, Nov 04 2015
%F A004331 D-finite with recurrence 3*(n-1)*(3*n-1)*(3*n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - _R. J. Mathar_, Mar 19 2025
%p A004331 #A004331
%p A004331 seq(binomial(4*n - 1,n), n = 0..20);
%t A004331 a[n_] := Binomial[4*n, n - 1]; Array[a, 19] (* _Amiram Eldar_, May 09 2020 *)
%o A004331 (PARI) vector(30, n, binomial(4*n, n-1)) \\ _Altug Alkan_, Nov 05 2015
%Y A004331 Cf. A002293, A005810, A052203, A224274, A257633, A262977.
%Y A004331 Cf. A001791, A004319, A004343, A004356, A004369, A004382.
%K A004331 nonn,easy
%O A004331 1,2
%A A004331 _N. J. A. Sloane_