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A372506 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-2*x) )^n.

%I A372506 #17 Oct 24 2024 05:19:52
%S A372506 1,3,23,198,1795,16758,159446,1537308,14967843,146833830,1449054178,
%T A372506 14369723316,143072565454,1429331585724,14320668653580,
%U A372506 143838879376248,1447883909314851,14602334949928710,147518977428892010,1492559101878005700,15121898521185194970
%N A372506 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-2*x) )^n.
%F A372506 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(3*n-1,n-k).
%F A372506 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x) * (1-2*x) ).
%F A372506 a(n) ~ (1 + sqrt(3)) * 2^(n - 3/2) * 3^((3*n-1)/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, May 04 2024
%F A372506 D-finite with recurrence 5*n*(n-1)*a(n) +18*(n-1)*(n-3)*a(n-1) +12*(-45*n^2+90*n-22)*a(n-2) -216*(3*n-7)*(3*n-8)*a(n-3)=0. - _R. J. Mathar_, Oct 24 2024
%p A372506 A372506 := proc(n)
%p A372506     add(binomial(n+k-1,k)*binomial(3*n-1,n-k),k=0..n) ;
%p A372506 end proc:
%p A372506 seq(A372506(n),n=0..80) ; # _R. J. Mathar_, Oct 24 2024
%t A372506 Table[SeriesCoefficient[1/((1 - x)*(1 - 2*x))^n, {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, May 04 2024 *)
%t A372506 Table[Binomial[3*n - 1, n] * Hypergeometric2F1[-n, n, 2*n, -1], {n, 0, 20}] (* _Vaclav Kotesovec_, May 04 2024 *)
%o A372506 (PARI) a(n, s=1, t=1, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t+u+1)*n-(s-1)*k-1, n-s*k));
%Y A372506 Cf. A001700, A069723, A085614, A255688, A372233.
%K A372506 nonn
%O A372506 0,2
%A A372506 _Seiichi Manyama_, May 04 2024