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%I #19 Jan 27 2019 07:31:17
%S 1,2,6,74,2942,379502,155417946,200991082378,814134608643518,
%T 10305926982053248142,406157795399324680023006,
%U 49758289996116571598723737976,18917910771770463473290738891259546,22290399373603219140501180230536732389992
%N a(n) = Sum_{k=0..n} binomial(n^2, k*(n-k)).
%C Ignoring initial term a(0), equals the logarithmic derivative of A207135.
%C Equals the row sums of triangle A228836.
%H Seiichi Manyama, <a href="/A207136/b207136.txt">Table of n, a(n) for n = 0..64</a>
%F a(n) ~ c * 2*sqrt(2/(3*Pi)) * (4/3^(3/4))^(n^2)/n, where c = EllipticTheta[3,0,1/3] = JacobiTheta3(0,1/3) = 1.69145968168171534... if n is even, and c = EllipticTheta[2,0,1/3] = JacobiTheta2(0,1/3) = 1.690611203075214233... if n is odd. - _Vaclav Kotesovec_, Mar 03 2014
%e L.g.f.: L(x) = 2*x + 6*x^2/2 + 74*x^3/3 + 2942*x^4/4 + 379502*x^5/5 +...
%e where exponentiation equals the g.f. of A207135:
%e exp(L(x)) = 1 + 2*x + 5*x^2 + 32*x^3 + 796*x^4 + 77508*x^5 +...
%e By definition, the initial terms begin: a(0) = 1;
%e a(1) = C(1,0) + C(1,0);
%e a(2) = C(4,0) + C(4,1) + C(4,0);
%e a(3) = C(9,0) + C(9,2) + C(9,2) + C(9,0);
%e a(4) = C(16,0) + C(16,3) + C(16,4) + C(16,3) + C(16,0);
%e a(5) = C(25,0) + C(25,4) + C(25,6) + C(25,6) + C(25,4) + C(25,0);
%e a(6) = C(36,0) + C(36,5) + C(36,8) + C(36,9) + C(36,8) + C(36,5) + C(36,0); ...
%e which is evaluated as:
%e a(1) = 1 + 1 = 2;
%e a(2) = 1 + 4 + 1 = 6;
%e a(3) = 1 + 36 + 36 + 1 = 74;
%e a(4) = 1 + 560 + 1820 + 560 + 1 = 2942;
%e a(5) = 1 + 12650 + 177100 + 177100 + 12650 + 1 = 379502;
%e a(6) = 1 + 376992 + 30260340 + 94143280 + 30260340 + 376992 + 1 = 155417946; ...
%p A207136:=n->add(binomial(n^2, k*(n-k)), k=0..n): seq(A207136(n), n=0..15); # _Wesley Ivan Hurt_, Jun 23 2015
%t Table[Sum[Binomial[n^2, k*(n-k)],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Mar 03 2014 *)
%o (PARI) {a(n)=sum(k=0,n,binomial(n^2,(n-k)*k))}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A207135 (exp), A167009, A228836.
%K nonn,nice,easy
%O 0,2
%A _Paul D. Hanna_, Feb 15 2012
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