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%I #8 Aug 23 2020 06:33:10
%S 1,8,7,128,304,177,3072,13952,21080,10199,98304,724992,2016000,
%T 2441056,1051745,3932160,42762240,187643904,407505664,428605352,
%U 169913511,188743680,2839019520,17974591488,60428242944,111985428352
%N Coefficients of polynomials S(n,x) related to Springer numbers.
%H A. Randrianarivony and J. Zeng, <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">Une famille des polynomes qui interpole plusieurs suites...</a>, Adv. Appl. Math. 17 (1996), 1-26.
%F Recurrence: S(0, x)=1, S(n, x)=(2x+2)(2x+4)S(n-1, x+2)-(2x+1)^2S(n-1, x).
%F G.f.: Sum[n>=0, S(n, x)t^n] = 1/(1+t-4*2(x+1)t/(1-4*2(x+2)t/(1+t-4*4(x+3)t/(1-4+4(x+4)t/...)))).
%e S(0,x) = 1,
%e S(1,x) = 8*x + 7,
%e S(2,x) = 128*x^2 + 304*x + 177,
%e S(3,x) = 3072*x^3 + 13952*x^2 + 21080*x + 10199.
%o (PARI) S(n,x)=if(n<1,1,(2*x+2)*(2*x+4)*S(n-1,x+2)-(2*x+1)^2*S(n-1,x))
%Y Cf. A001586. S(n, 1/2) = A000464(n+1), S(n, -1/2) = A000281(n).
%Y Leading coefficients are A051189. Constant terms are in A098433.
%Y Cf. A001586. S(n, 1/2) = A000464(n), S(n, -1/2) = A000281(n).
%K tabl,nonn
%O 0,2
%A _Ralf Stephan_, Sep 07 2004
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