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A266488 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (x/N)^n/n! ]^(1/N).

%I A266488 #5 Dec 30 2015 18:14:48
%S A266488 1,1,0,1,4,0,1,12,42,0,1,24,216,752,0,1,40,660,5440,19360,0,1,60,1560,
%T A266488 22320,178920,654912,0,1,84,3150,68320,916440,7316064,27546736,0,1,
%U A266488 112,5712,173600,3432800,44693376,359051392,1388207872,0,1,144,9576,387072,10493280,197261568,2536797312,20605529088,81621893376,0,1,180,15120,782880,27735120,702777600,12845683200,164732083200,1355581612800,5488951731200,0
%N A266488 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (x/N)^n/n! ]^(1/N).
%C A266488 Compare to: Limit_{N->oo} [ Sum_{n>=0} (N + n)^n * x^n/n! ]^(1/N) = Sum_{n>=0} (n+1)^(n-1) * x^n/n!.
%e A266488 E.g.f. A(x) = 1 + x +
%e A266488 x^2/2! * (1 + 4*y) +
%e A266488 x^3/3! * (1 + 12*y + 42*y^2) +
%e A266488 x^4/4! * (1 + 24*y + 216*y^2 + 752*y^3) +
%e A266488 x^5/5! * (1 + 40*y + 660*y^2 + 5440*y^3 + 19360*y^4) +
%e A266488 x^6/6! * (1 + 60*y + 1560*y^2 + 22320*y^3 + 178920*y^4 + 654912*y^5) +
%e A266488 x^7/7! * (1 + 84*y + 3150*y^2 + 68320*y^3 + 916440*y^4 + 7316064*y^5 + 27546736*y^6) +
%e A266488 x^8/8! * (1 + 112*y + 5712*y^2 + 173600*y^3 + 3432800*y^4 + 44693376*y^5 + 359051392*y^6 + 1388207872*y^7) + ...
%e A266488 where A(x) equals the limit, as N -> oo, of the series
%e A266488 [1 + (N + y)^2*(x/N) + (N + 2*y)^4*(x/N)^2/2! + (N + 3*y)^6*(x/N)^3/3! + (N + 4*y)^8*(x/N)^4/4! + (N + 5*y)^10*(x/N)^5/5! + (N + 6*y)^12*(x/N)^6/6! +...]^(1/N).
%e A266488 Triangle of coefficients T(n,k) of x^n*y^k/n!, n>=0, k=0..n, begins:
%e A266488 1;
%e A266488 1, 0;
%e A266488 1, 4, 0;
%e A266488 1, 12, 42, 0;
%e A266488 1, 24, 216, 752, 0;
%e A266488 1, 40, 660, 5440, 19360, 0;
%e A266488 1, 60, 1560, 22320, 178920, 654912, 0;
%e A266488 1, 84, 3150, 68320, 916440, 7316064, 27546736, 0;
%e A266488 1, 112, 5712, 173600, 3432800, 44693376, 359051392, 1388207872, 0;
%e A266488 1, 144, 9576, 387072, 10493280, 197261568, 2536797312, 20605529088, 81621893376, 0;
%e A266488 1, 180, 15120, 782880, 27735120, 702777600, 12845683200, 164732083200, 1355581612800, 5488951731200, 0;
%e A266488 1, 220, 22770, 1467840, 65659440, 2143842624, 52117998240, 938463651840, 12065358919680, 100649306644480, 415721105434624, 0; ...
%o A266488 (PARI) /* Print the initial rows of this triangle: */
%o A266488 \p400
%o A266488 P(n) = sum(k=0, 21, (n + k*y)^(2*k) * (x/n)^k/k! +O(x^21))
%o A266488 V=Vec( round( serlaplace( P(10^100)^(1/10^100) )*1.) )
%o A266488 for(n=1,15,print(Vec(V[n]+O(y^n))))
%Y A266488 Cf. A266481, A266482, A266483, A266484, A266485, A266486, A266487.
%K A266488 nonn
%O A266488 0,5
%A A266488 _Paul D. Hanna_, Dec 30 2015