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%I #20 May 31 2020 21:58:14
%S 1,0,-3,28,-255,2376,-20195,71688,3834369,-187855280,6676401501,
%T -220595216280,7180102389889,-234023553073296,7631745228481725,
%U -245429882267144624,7501602903392006145,-196609711096827812448,2542435002501531333949
%N Coefficient of x^n in (1 - (n-1)*x - n*x^2)^n.
%C Also coefficient of x^n in the expansion of 1/sqrt(1 + 2*(n-1)*x + ((n+1)*x)^2).
%H Seiichi Manyama, <a href="/A307885/b307885.txt">Table of n, a(n) for n = 0..386</a>
%F a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^2.
%F a(n) = Sum_{k=0..n} (-n-1)^(n-k) * binomial(n,k) * binomial(n+k,k).
%F a(n) = Hypergeometric2F1(-n, -n, 1, -n). - _Vaclav Kotesovec_, May 07 2019
%F a(n) = n! * [x^n] exp((1 - n)*x) * BesselI(0,2*sqrt(-n)*x). - _Ilya Gutkovskiy_, May 31 2020
%p A307885:= n -> simplify(hypergeom([-n,-n], [1], -n));
%p seq(A307885(n), n = 0..30); # _G. C. Greubel_, May 31 2020
%t Table[Hypergeometric2F1[-n, -n, 1, -n], {n, 0, 20}] (* _Vaclav Kotesovec_, May 07 2019 *)
%o (PARI) {a(n) = polcoef((1-(n-1)*x-n*x^2)^n, n)}
%o (PARI) {a(n) = sum(k=0, n, (-n)^k*binomial(n, k)^2)}
%o (PARI) {a(n) = sum(k=0, n, (-n-1)^(n-k)*binomial(n, k)*binomial(n+k, k))}
%o (Sage) [ hypergeometric([-n, -n], [1], -n).simplify_hypergeometric() for n in (0..30)] # _G. C. Greubel_, May 31 2020
%Y Main diagonal of A307884.
%Y Cf. A187021.
%K sign
%O 0,3
%A _Seiichi Manyama_, May 02 2019
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