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%I #18 Mar 09 2020 12:34:31
%S 1,8,95,1336,20642,338640,5791291,102108760,1842857390,33879118384,
%T 632210693270,11944142806064,228010741228740,4391334026631072,
%U 85221618348230355,1664901954576830360,32716286416687895862,646228961799752926320,12823701194384778672322
%N O.g.f. exp( Sum_{n >= 1} A262732(n)*x^n/n ).
%C O.g.f. is 1/x * the series reversion of x*(1 - x)^k/(1 + x)^(k+2) for k = 3. See the cross references for related sequences obtained from other values of k.
%H Peter Bala, <a href="/A100100/a100100_1.pdf">Notes on logarithmic differentiation, the binomial transform and series reversion</a>
%F a(n-1) = 1/n * Sum_{i = 0..n-1} binomial(5*n,i)*binomial(4*n-i-2,n-i-1).
%F O.g.f.: A(x) = exp ( Sum_{n >= 1} 1/n! * (5*n)!/(5*n/2)! * (3*n/2)!/(3*n)!*x^n/n ) = 1 + 8*x + 195*x^2 + 1336*x^3 + ....
%F 1 + x*A'(x)/A(x) is the o.g.f. for A262732.
%F O.g.f. is the series reversion of x*(1 - x)^3/(1 + x)^5.
%F a(0) = 1 and for n >= 1, a(n) = 1/n * Sum {k = 1..n} 1/k!*(5*k)!/GAMMA(5*k/2+1)*GAMMA(3*k/2+1)/(3*k)! * a(n-k).
%p A262737 := proc (n) option remember; if n = 0 then 1 else add(1/k!*(5*k)!/GAMMA(5*k/2 + 1)*GAMMA(3*k/2 + 1)/(3*k)!*A262737(n-k), k = 1 .. n)/n end if; end proc:
%p seq(A262737(n), n = 0 .. 20);
%o (PARI) a(n) = sum(k=0, n, binomial(5*(n+1),k)*binomial(4*(n+1)-k-2,(n+1)-k-1))/(n+1); \\ _Altug Alkan_, Oct 03 2015
%Y Cf. A000108 (k = 0), A007297 (k = 1), A066357 (k = 2), A262738 (k = 4), A262739 (k = 5), A262740 (k = 6), A262732.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Sep 29 2015
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