%I #32 Jun 05 2017 19:07:09

%S 1,1,5,73,2017,86801,5289301,430814665,45052534913,5868875082817,

%T 930114039075301,175964489469769001,39125942325820605025,

%U 10092849114680961297553,2987365449592984040715317,1005030253302269078318250601

%N Square row sums of the triangle of Lah numbers (A105278).

%H G. C. Greubel, <a href="/A276965/b276965.txt">Table of n, a(n) for n = 0..245</a>

%F a(n) = Sum_{k=0..n} lah(n,k)^2.

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

%F a(n) = hypergeometric([-n+1,-n+1,-n,-n],[1],1).

%F a(n) = (n!)^2 * hypergeometric([-n+1,-n+1],[1,2,2],1) for n > 0.

%F Recurrence: n*(16*n^3 - 96*n^2 + 185*n - 116)*a(n) = 2*(32*n^6 - 272*n^5 + 930*n^4 - 1668*n^3 + 1670*n^2 - 867*n + 164)*a(n-1) - (n-2)*(96*n^7 - 1056*n^6 + 4646*n^5 - 10500*n^4 + 12990*n^3 - 8644*n^2 + 2827*n - 364)*a(n-2) + 2*(n-3)*(n-2)^3*(32*n^6 - 336*n^5 + 1410*n^4 - 2978*n^3 + 3268*n^2 - 1731*n + 353)*a(n-3) - (n-4)^2*(n-3)^3*(n-2)^4*(16*n^3 - 48*n^2 + 41*n - 11)*a(n-4). - _Vaclav Kotesovec_, Sep 27 2016

%F a(n) ~ n^(2*n - 3/4) * exp(4*sqrt(n) - 2*n - 1) / (2^(3/2) * sqrt(Pi)) * (1 + 31/(96*sqrt(n)) + 937/(18432*n)). - _Vaclav Kotesovec_, Sep 27 2016

%t Table[HypergeometricPFQ[{1-n,1-n,-n,-n},{1},1],{n,0,100}]

%o (Maxima) makelist(hypergeometric([-n+1,-n+1,-n,-n],[1],1),n,0,12);

%o (Perl) use ntheory ":all"; for my $n (0..20) { say "$n ",vecsum(map{my $l=stirling($n,$_,3); vecprod($l,$l); } 0..$n) } # _Dana Jacobsen_, Mar 16 2017

%o (PARI) concat([1], for(n=1,25, print1(sum(k=0,n, binomial(n,k)^2*binomial(n-1,k-1)^2*((n-k)!)^2), ", "))) \\ _G. C. Greubel_, Jun 05 2017

%Y Cf. A000262, A105278, A008297.

%K nonn

%O 0,3

%A _Emanuele Munarini_, Sep 27 2016