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A099169 a(n) = (1/n) * Sum_{k=0..n-1} C(n,k) * C(n,k+1) * (n-1)^k.

%I A099169 #28 Dec 01 2021 15:57:43
%S A099169 1,2,11,100,1257,20076,387739,8766248,226739489,6595646860,
%T A099169 212944033051,7550600079672,291527929539433,12169325847587832,
%U A099169 545918747361417291,26183626498897556176,1336713063706757646465
%N A099169 a(n) = (1/n) * Sum_{k=0..n-1} C(n,k) * C(n,k+1) * (n-1)^k.
%C A099169 A diagonal of Narayana array (A008550).
%H A099169 Vincenzo Librandi, <a href="/A099169/b099169.txt">Table of n, a(n) for n = 1..100</a>
%H A099169 Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
%F A099169 From _Vaclav Kotesovec_, Apr 18 2014, extended Dec 01 2021: (Start)
%F A099169 a(n) = Hypergeometric2F1([1-n,-n], [2], -1+n).
%F A099169 a(n) ~ exp(2*sqrt(n)-2) * n^(n-7/4) / (2*sqrt(Pi)) * (1 + 119/(48*sqrt(n))). (End)
%p A099169 A099169:= n-> add( binomial(n, j)*binomial(n-1,j)*(n-1)^j/(j+1), j=0..n-1);
%p A099169 seq( A099169(n), n=1..30) # _G. C. Greubel_, Feb 16 2021
%t A099169 Join[{1},Table[Sum[Binomial[n,k]Binomial[n,k+1](n-1)^k,{k,0,n-1}]/n,{n,2,20}]] (* _Harvey P. Dale_, Oct 07 2013 *)
%t A099169 Table[Hypergeometric2F1[1-n,-n,2,-1+n],{n,1,20}] (* _Vaclav Kotesovec_, Apr 18 2014 *)
%o A099169 (Sage)
%o A099169 def A099169(n): return sum( binomial(n, j)*binomial(n-1,j)*(n-1)^j/(j+1) for j in [0..n-1])
%o A099169 [A099169(n) for n in [1..30]] # _G. C. Greubel_, Feb 16 2021
%o A099169 (Magma)
%o A099169 A099169:= func< n | (&+[Binomial(n, j)*Binomial(n-1,j)*(n-1)^j/(j+1): j in [0..n-1]]) >;
%o A099169 [A099169(n): n in [1..30]]; // _G. C. Greubel_, Feb 16 2021
%o A099169 (PARI) a(n) = (1/n) * sum(k=0, n-1, binomial(n,k) * binomial(n,k+1) * (n-1)^k); \\ _Michel Marcus_, Feb 16 2021
%Y A099169 Cf. A092366, A187018, A187019, A187021.
%Y A099169 Cf. A008550, A204057, A243631.
%K A099169 nonn,easy
%O A099169 1,2
%A A099169 _Ralf Stephan_, Oct 09 2004