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A360101 a(n) = Sum_{k=0..n} binomial(n+4*k-1,n-k) * Catalan(k).

%I A360101 #20 Aug 16 2023 08:21:55
%S A360101 1,1,7,40,234,1432,9078,59113,393125,2659233,18240801,126588424,
%T A360101 887221916,6271153060,44652824248,319990906290,2306133322704,
%U A360101 16703784324239,121534039921585,887845073567240,6509750423778460,47888814944642434,353362258550740732
%N A360101 a(n) = Sum_{k=0..n} binomial(n+4*k-1,n-k) * Catalan(k).
%F A360101 G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^5.
%F A360101 G.f.: c(x/(1-x)^5), where c(x) is the g.f. of A000108.
%F A360101 D-finite with recurrence (n+1)*a(n) +(-10*n+7)*a(n-1) +(19*n-56)*a(n-2) +10*(-2*n+9)*a(n-3) +5*(3*n-19)*a(n-4) +(-6*n+49)*a(n-5) +(n-10)*a(n-6)=0. - _R. J. Mathar_, Mar 12 2023
%p A360101 A360101 := proc(n)
%p A360101     add(binomial(n+4*k-1,n-k)*A000108(k),k=0..n) ;
%p A360101 end proc:
%p A360101 seq(A360101(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023
%t A360101 m = 23;
%t A360101 A[_] = 0;
%t A360101 Do[A[x_] = 1 + x A[x]^2/(1 - x)^5 + O[x]^m // Normal, {m}];
%t A360101 CoefficientList[A[x], x] (* _Jean-François Alcover_, Aug 16 2023 *)
%o A360101 (PARI) a(n) = sum(k=0, n, binomial(n+4*k-1, n-k)*binomial(2*k, k)/(k+1));
%o A360101 (PARI) my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x)^5)))
%Y A360101 Partial sums are A360103.
%Y A360101 Cf. A000108, A360057.
%Y A360101 Cf. A002212, A006319, A162475, A360100.
%K A360101 nonn
%O A360101 0,3
%A A360101 _Seiichi Manyama_, Jan 25 2023