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%I #18 Jul 20 2020 02:05:14
%S 1,12,215,4564,106442,2635704,68031147,1810302340,49308457334,
%T 1368019979976,38525145673126,1098380420669000,31641932951483220,
%U 919622628946689648,26931762975278938035,793967020231145502564,23543663463050594677310,701763102761640853890600,21014048069544552257072530,631868353403527700756671320,19070677448561228207945931276
%N O.g.f. exp( Sum_{n >= 1} A262733(n)*x^n/n ).
%C O.g.f. is 1/x * the series reversion of x*(1 - x)^k/(1 + x)^(k+2) when k = 5. 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(7*n,i)*binomial(6*n-i-2,n-i-1).
%F O.g.f.: A(x) = exp ( Sum_{n >= 1} 1/n! * (7*n)!/(7*n/2)! * (5*n/2)!/(5*n)!*x^n/n ) = 1 + 12*x + 215*x^2 + 4564*x^3 + ....
%F 1 + x*A'(x)/A(x) is the o.g.f. for A262733.
%F O.g.f. is the series reversion of x*(1 - x)^5/(1 + x)^7,
%F a(0) = 1 and for n >= 1, a(n) = 1/n * Sum {k = 1..n} 1/k! * (7*k)!/(7*k/2)! * (5*k/2)!/(5*k)!*a(n-k).
%p A262739 := proc (n) option remember; if n = 0 then 1 else add(1/k!*(7*k)!/GAMMA(7*k/2 + 1)*GAMMA(5*k/2 + 1)/(5*k)!*A262739(n-k), k = 1 .. n)/n end if; end proc:
%p seq(A262739(n), n = 0..20);
%o (PARI) a(n) = sum(k=0, n, binomial(7*(n+1),k)*binomial(6*(n+1)-k-2,(n+1)-k-1))/(n+1); \\ _Altug Alkan_, Oct 03 2015
%Y Cf. A262733, A211419, A000108 (k = 0), A007297 (k = 1), A066357 (k = 2), A262737 (k = 3), A262738 (k = 4), A262740 (k = 6).
%K nonn,easy
%O 0,2
%A _Peter Bala_, Sep 29 2015
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