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A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

%I A167009 #42 Aug 30 2022 14:15:39
%S A167009 1,2,8,170,16512,6643782,11582386286,79450506979090,
%T A167009 2334899414608412672,265166261617029717011822,
%U A167009 128442558588779813655233443038,238431997806538515396060130910954852
%N A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).
%H A167009 Vincenzo Librandi, <a href="/A167009/b167009.txt">Table of n, a(n) for n = 0..58</a>
%H A167009 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Interesting asymptotic formulas for binomial sums</a>, Jun 09 2013.
%F A167009 Ignoring initial term, equals the logarithmic derivative of A167006. - _Paul D. Hanna_, Nov 18 2009
%F A167009 If n is even then a(n) ~ c * 2^(n^2 + 1/2)/(n*sqrt(Pi)), where c = Sum_{k = -infinity..infinity} exp(-2*k^2) = 1.271341522189... (see A218792). - _Vaclav Kotesovec_, Nov 05 2012
%F A167009 If n is odd then c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... - _Vaclav Kotesovec_, Nov 06 2012
%F A167009 a(n) = A306846(n^2,n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^n) for n > 0. - _Seiichi Manyama_, Oct 11 2021
%e A167009 The triangle A209330 of coefficients C(n^2, n*k), n>=k>=0, begins:
%e A167009   1;
%e A167009   1,       1;
%e A167009   1,       6,          1;
%e A167009   1,      84,         84,          1;
%e A167009   1,    1820,      12870,       1820,          1;
%e A167009   1,   53130,    3268760,    3268760,      53130,       1;
%e A167009   1, 1947792, 1251677700, 9075135300, 1251677700, 1947792,     1; ...
%e A167009 in which the row sums form this sequence.
%t A167009 Table[Sum[Binomial[n^2,n*k],{k,0,n}],{n,0,15}] (* _Harvey P. Dale_, Dec 11 2011 *)
%o A167009 (PARI) a(n)=sum(k=0,n,binomial(n^2,n*k))
%o A167009 (Magma) [(&+[Binomial(n^2, n*j): j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 26 2022
%o A167009 (Sage) [sum(binomial(n^2, n*j) for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Aug 26 2022
%Y A167009 Cf. A014062, A167006, A167010, A209330, A218792, A306846.
%K A167009 nonn
%O A167009 0,2
%A A167009 _Paul D. Hanna_, Nov 17 2009