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A202474 Coefficients of y^(n-1) in Product_{k=1..n} (1 + k*y + y^2) for n >= 1.

%I A202474 #17 Mar 02 2019 15:06:16
%S A202474 1,3,14,80,539,4179,36630,358056,3860922,45519870,582466235,
%T A202474 8038684290,119018991779,1881685721265,31638175704546,563703015007056,
%U A202474 10609073237333432,210305960538762456,4379808881917047898,95604092878386437940,2182706554812339958778
%N A202474 Coefficients of y^(n-1) in Product_{k=1..n} (1 + k*y + y^2) for n >= 1.
%F A202474 E.g.f.: 1/(1-x) * Sum_{n>=0} log(1 - x)^(2*n+1) / (n!*(n+1)!).
%e A202474 E.g.f.: A(x) = x + 3*x^2/2! + 14*x^3/3! + 80*x^4/4! + 539*x^5/5! + 4179*x^6/6! + 36630*x^7/7! + 358056*x^8/8! + 3860922*x^9/9! + 45519870*x^10/10! + ...
%e A202474 The coefficients in Product_{k=1..n} (1+k*x+x^2), n>=0, form the triangle:
%e A202474 [1];
%e A202474 [(1), 1, 1];
%e A202474 [1,(3), 4, 3, 1];
%e A202474 [1, 6, (14), 18, 14, 6, 1];
%e A202474 [1, 10, 39, (80), 100, 80, 39, 10, 1];
%e A202474 [1, 15, 90, 285, (539), 660, 539, 285, 90, 15, 1];
%e A202474 [1, 21, 181, 840, 2339, (4179), 5038, 4179, 2339, 840, 181, 21, 1];
%e A202474 [1, 28, 329, 2128, 8400, 21392, (36630), 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ...
%e A202474 the coefficients in parenthesis form the initial terms of this sequence.
%t A202474 Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,1,n+1}]],x^n],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 10 2015 *)
%o A202474 (PARI) {a(n) = polcoeff(prod(k=1,n,1 + k*x + x^2 +x*O(x^n)),n-1)}
%o A202474 for(n=1,30,print1(a(n),", "))
%Y A202474 Cf. A201826, A201950, A202476.
%K A202474 nonn
%O A202474 1,2
%A A202474 _Paul D. Hanna_, Dec 19 2011
%E A202474 Changed offset to 1 to agree with e.g.f. - _Paul D. Hanna_, Mar 02 2019