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A181067 a(n) = Sum_{k=0..n-1} binomial(n-1,k)^2 * binomial(n,k).

%I A181067 #29 Apr 06 2021 04:49:17
%S A181067 1,3,16,95,606,4032,27616,193167,1372930,9881498,71846160,526764680,
%T A181067 3889340560,28888634400,215680108416,1617467908751,12177754012458,
%U A181067 92004463332486,697263463622080,5298985086555090,40371796982444356
%N A181067 a(n) = Sum_{k=0..n-1} binomial(n-1,k)^2 * binomial(n,k).
%H A181067 Vincenzo Librandi, <a href="/A181067/b181067.txt">Table of n, a(n) for n = 1..200</a>
%F A181067 L.g.f.: Sum_{n>=1} [ Sum_{k>=0} binomial(n+k-1,k)^3 *x^k ] *x^n/n.
%F A181067 Logarithmic derivative of A181066.
%F A181067 Recurrence: n^2*a(n) = - (n^2-17*n+10)*a(n-1) + 48*(n^2-3*n+1)*a(n-2) + 16*(n-3)*(11*n-36)*a(n-3) + 128*(n-4)^2*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F A181067 a(n) ~ sqrt(3)*8^n/(6*Pi*n). - _Vaclav Kotesovec_, Oct 24 2012
%F A181067 a(n) = 3F2([1-n, 1-n, -n], [1, 1], -1). - _Pierre-Louis Giscard_, Jul 20 2013
%F A181067 a(n) = n * hypergeometric([-n+1,-n+1,-n+1], [1,2], -1) for n > 0. - _Emanuele Munarini_, Sep 27 2016
%F A181067 a(n) = Sum_{k=0..n-1} ((n-k)/n)^2 * binomial(n,k)^3. - _G. C. Greubel_, Apr 05 2021
%e A181067 L.g.f.: L(x) = x + 3*x^2/2 + 16*x^3/3 + 95*x^4/4 + 606*x^5/5 + ...
%e A181067 which equals the series:
%e A181067   L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...)*x
%e A181067   + (1 + 2^3*x +  3^3*x^2 +  4^3*x^3 +   5^3*x^4 +   6^3*x^5 + ...)*x^2/2
%e A181067   + (1 + 3^3*x +  6^3*x^2 + 10^3*x^3 +  15^3*x^4 +  21^3*x^5 + ...)*x^3/3
%e A181067   + (1 + 4^3*x + 10^3*x^2 + 20^3*x^3 +  35^3*x^4 +  56^3*x^5 + ...)*x^4/4
%e A181067   + (1 + 5^3*x + 15^3*x^2 + 35^3*x^3 +  70^3*x^4 + 126^3*x^5 + ...)*x^5/5
%e A181067   + (1 + 6^3*x + 21^3*x^2 + 56^3*x^3 + 126^3*x^4 + 252^3*x^5 + ...)*x^6/6 + ...
%e A181067 Exponentiation yields the g.f. of A181066:
%e A181067   exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 157*x^5 + 865*x^6 + ... + A181066(n)*x^n + ...
%p A181067 A181067:= n-> add(((n-k)/n)^2*binomial(n,k)^3, k=0..n-1); seq(A181067(n), n=1..25); # _G. C. Greubel_, Apr 05 2021
%t A181067 Table[Sum[Binomial[n-1,k]^2*Binomial[n,k],{k,0,n-1}], {n,1,20}] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%t A181067 Table[HypergeometricPFQ[{1-n, 1-n, -n}, {1, 1}, -1], {n,1,20}] (* _Pierre-Louis Giscard_, Jul 20 2013 *)
%o A181067 (PARI) {a(n)=sum(k=0, n-1, binomial(n-1, k)^3*n/(n-k))}
%o A181067 (PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^3*x^k)*x^m/m)+x*O(x^n), n)}
%o A181067 (Maxima) makelist(hypergeometric([-n+1,-n+1,-n],[1,1],-1),n,0,12); /* _Emanuele Munarini_, Sep 27 2016 */
%o A181067 (Magma) [(&+[ ((n-k)/n)^2*Binomial(n,k)^3 : k in [0..n-1]]): n in [1..25]]; // _G. C. Greubel_, Apr 05 2021
%o A181067 (Sage) [sum( ((n-k)/n)^2*binomial(n,k)^3 for k in (0..n-1) ) for n in (1..25)] # _G. C. Greubel_, Apr 05 2021
%Y A181067 Cf. A181066 (exp), A181069 (variant).
%K A181067 nonn
%O A181067 1,2
%A A181067 _Paul D. Hanna_, Oct 03 2010