A262740 - OEIS

%I #13 Mar 09 2020 12:35:05

%S 1,14,293,7266,197962,5726364,172662765,5367187226,170772853790,

%T 5534640052292,182070248073826,6063785526898644,204055962203476788,

%U 6927718839334775608,236994877398511998717,8161492483543100398410,282705062046649346154006,9843330120848835962213940

%N O.g.f. exp( Sum_{n >= 1} A211421(n)*x^n/n ).

%C O.g.f. is 1/x * the series reversion of x*(1 - x)^k/(1 + x)^(k+2) at k = 6. 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(8*n,i)*binomial(7*n-i-2,n-i-1).

%F O.g.f.: A(x) = exp ( Sum_{n >= 1} 1/n! * (8*n)!/(4*n)! * (3*n)!/(6*n)!*x^n/n ) = 1 + 14*x + 293*x^2 + 7266*x^3 + ....

%F 1 + x*A'(x)/A(x) is the o.g.f. for A211421.

%F O.g.f. is the series reversion of x*(1 - x)^6/(1 + x)^8.

%F a(0) = 1 and for n >= 1, a(n) = 1/n * Sum {k = 1..n} 1/k! * (8*k)!/(4*k)! * (3*k)!/(6*k)!*a(n-k).

%p #A262740

%p A262740 := proc (n) option remember; if n = 0 then 1 else add(1/k!*(8*k)!/(4*k)!*(3*k)!/(6*k)!*A262740(n-k), k = 1 .. n)/n end if; end proc:

%p seq(A262740(n), n = 0..17);

%o (PARI) a(n) = sum(k=0, n, binomial(8*(n+1),k)*binomial(7*(n+1)-k-2,(n+1)-k-1))/(n+1); \\ _Altug Alkan_, Oct 03 2015

%Y Cf. A211421, A000108 (k = 0), A007297 (k = 1), A066357 (k = 2), A262737 (k = 3), A262738 (k = 4), A262739 (k = 5).

%K nonn,easy

%O 0,2

%A _Peter Bala_, Sep 29 2015