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%I #18 Mar 02 2019 15:01:29
%S 1,1,4,18,100,660,5038,43624,422252,4516380,52885644,672781824,
%T 9238314358,136175455234,2144494356834,35930786795040,638168940129732,
%U 11976278012219556,236791150694618872,4919643784275283480,107152493449339765396,2441410548192907949196
%N Central coefficients in Product_{k=1..n} (1 + k*y + y^2).
%F E.g.f.: 1/(1-x) * Sum_{n>=0} log(1 - x)^(2*n) / n!^2. - _Paul D. Hanna_, Mar 02 2019
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 100*x^4/4! + 660*x^5/5! + 5038*x^6/6! + 43624*x^7/7! + 422252*x^8/8! + 4516380*x^9/9! + 52885644*x^10/10! + ...
%e The coefficients in Product_{k=1..n} (1 + k*y + y^2), n>=0, form triangle A249790:
%e [1];
%e [1, 1, 1];
%e [1, 3, 4, 3, 1];
%e [1, 6, 14, 18, 14, 6, 1];
%e [1, 10, 39, 80, 100, 80, 39, 10, 1];
%e [1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1];
%e [1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1];
%e [1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ...
%e in which the central terms of the rows form this sequence.
%t Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,1,n}]],x^n],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 10 2015 *)
%o (PARI) {a(n) = polcoeff(prod(k=1,n, 1 + k*x + x^2 +x*O(x^n)), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = n!*polcoeff( sum(m=0, n, log(1 - x +x*O(x^n))^(2*m)/m!^2 ) / (1 - x +x*O(x^n)), n)}
%o for(n=0, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 02 2019
%Y Cf. A249790, A202474, A201950, A202476.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 19 2011
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