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A300870 E.g.f. A(x) satisfies: [x^n] A(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.

%I A300870 #7 Oct 14 2020 02:44:38
%S A300870 1,1,7,307,37537,8755561,3304572391,1847063377867,1447456397632897,
%T A300870 1532041772833285777,2130468278450240803591,3808068399270998260188451,
%U A300870 8590473242021318921848038817,24074336129439663228349612217977,82657249526888437632759608331784807,343425012928825298349935150449843384891,1707701025594135213863151839769061397729281
%N A300870 E.g.f. A(x) satisfies: [x^n] A(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.
%C A300870 Compare e.g.f. to: [x^n] exp(x)^(n*(n+1)) = (n+1) * [x^(n-1)] exp(x)^(n*(n+1)) for n>=1.
%H A300870 Paul D. Hanna, <a href="/A300870/b300870.txt">Table of n, a(n) for n = 0..200</a>
%F A300870 a(n) ~ c * n!^3 * n^3, where c = 0.044039511494832369374... - _Vaclav Kotesovec_, Oct 14 2020
%e A300870 E.g.f.: A(x) = 1 + x + 7*x^2/2! + 307*x^3/3! + 37537*x^4/4! + 8755561*x^5/5! + 3304572391*x^6/6! + 1847063377867*x^7/7! + 1447456397632897*x^8/8! + 1532041772833285777*x^9/9! + ...
%e A300870 ILLUSTRATION OF DEFINITION.
%e A300870 The table of coefficients of x^k in A(x)^(n*(n+1)) begins:
%e A300870 n=1: [(1), (2), 8, 328/3, 9728/3, 2241184/15, 420248704/45, ...];
%e A300870 n=2: [1, (6), (36), 432, 11328, 2470464/5, 150254784/5, ...];
%e A300870 n=3: [1, 12, (108), (1296), 29136, 5776128/5, 335166336/5, ...];
%e A300870 n=4: [1, 20, 260, (10480/3), (209600/3), 7265600/3, 1173400640/9, ...];
%e A300870 n=5: [1, 30, 540, 8640, (166800), (5004000), 241367040, 116509893120/7...];
%e A300870 n=6: [1, 42, 1008, 19656, 396816, (53339328/5), (2240251776/5), ...];
%e A300870 n=7: [1, 56, 1736, 124096/3, 2767184/3, 355355392/15, (38932329856/45), (2180210471936/45), ...]; ...
%e A300870 in which the coefficients in parenthesis are related by
%e A300870 2 = 1*2*(1); 36 = 2*3*(6); 1296 = 3*4*(108); 209600/3 = 4*5*(10480/3); 5004000 = 5*6*(166800); 2240251776/5 = 6*7*(53339328/5); ...
%e A300870 illustrating that: [x^n] A(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] A(x)^(n*(n+1)).
%e A300870 LOGARITHMIC PROPERTY.
%e A300870 The logarithm of the e.g.f. is the integer series:
%e A300870 log(A(x)) = x + 3*x^2 + 48*x^3 + 1510*x^4 + 71280*x^5 + 4511808*x^6 + 361640832*x^7 + 35516910960*x^8 + 4184770003200*x^9 + ... + A300871(n)*x^n + ...
%o A300870 (PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)*(#A))); A[#A] = ((#A-1)*(#A)*V[#A-1] - V[#A])/(#A-1)/(#A) ); EGF=Ser(A); n!*A[n+1]}
%o A300870 for(n=0, 20, print1(a(n), ", "))
%Y A300870 Cf. A300871, A300590, A296170, A182962.
%K A300870 nonn
%O A300870 0,3
%A A300870 _Paul D. Hanna_, Mar 14 2018